Training horses always involves some risk of injury (or even death) to the horse, the trainer, any spectators, equipment used, etc. - this risk is totally your own as you will be making your own judgements on what to do. As with any suggestions on any subject you must weigh them out for yourself and proceed accordingly - at your own risk.
Many of the problems humans have with horses stem, in my opinion, from the fact that humans (even the vegetarians) see and react to the world through the eyes of a predator while horses see and react to the world through the eyes of a prey animal.
If you have ever watched predators (even your house cat) you will note they have their eyes on the front rather than the sides of their head and they tend to take a direct route toward where they are going - particularly so if they have locked onto some target. By contrast prey animals spend much of their time grazing or browsing perhaps with some destination in mind but with a lot less direct route involved getting there OR even if they take a direct route it often includes lots of starts and stops to sample tasty food items along the way. So what does that have to do with getting on your green horse safely the very first time? Well if you want to avoid having problems with horses you need to first realize you move like a predator - you home in on your target and take the most direct route between you and that target. So what? Well the horse, being a prey animal, reads your movement as "a predator is coming" and its first line of defense is to put distance between itself and any predatory threat.
Watch a house cat trying to catch a rabbit, bird, or squirrel and you will note that predators with a chosen target (if they start fairly far away) tend to close distance as rapidly as they think they can get away with until they get just outside of the prey animals "flee in panic" zone. At that point they slow way down - and try to disappear into the ground - closing the distance slowly. Then if/when they get close enough they suddenly burst into action to pounce on their prey.
Hmmm, that sorta sounds like the sequence followed when mounting a horse doesn't it? You close the distance between you and the horse fairly rapidly, once you get close you slow way down, then you suddenly spring upward onto their back.
Do you think maybe by doing that we might be triggering a very natural, instinctive, anti-predator response?
Now if you were a prey animal and you suddenly felt or saw a predator on your back what would you do? Probably anything you could to get it off - like "swallowing your head" and bucking.
From observation and decades of experience combined with a bit of early-on "learning the hard way" I have come to the conclusion that first time mounting has the potential to trip four different anti-predator "triggers." And while a prey animal rarely reacts to a single anti-predator trigger they do react to two or more anti-predator triggers that get pulled at the same time.
The four anti-predator triggers you have the potential to trip when first mounting a horse are: rapid upward motion close to prey; pouncing on back; squeezing pressure around barrel (picture a mountain lion trying to hang on); something higher than horse's head on back.
Generally speaking when people work slow and easy and basically smooze the horse into allowing first saddling - then mount up right away but also real easy - then they most generally get away with that first brief easy "smooze your way through it" ride. They walk around a bit, get off and call it a miracle. Then some other early ride - often the very next ride they let their confidence get the best of them and they push the youngling to trot. The youngster is unbalanced under all that weight on its back (picture yourself carrying a heavy backpack for the first time) and the rider gets slightly bounced out of the saddle and when they touch back down the horse translates it as a "pounce on back" with possibly the girth already giving the "hold on squeeze" and the rider being up above horses head on back --- 3 triggers pulled and it only takes two. This is a recipe for a rodeo event - saddle bronc. And it can be, in my opinion, completely avoided if you just "start right."
I do realize there are MANY ways to get a young horse started. I have seen people succeed in a variety of ways - and so have I. I know that some advocate intentionally allowing a young saddled horse to run free and bucking with just the saddle on to "get it out of them" because they can't get rid of that saddle. And I do know it can be done myriad ways. I just tend to believe that if my objective is to end up with a horse that NEVER bucks then maybe I should start it out in a way that doesn't involve the horse ever bucking with a human or even just a saddle on its back.
So can that really be done? Can you start out a completely green horse so that it NEVER bucks with a human on its back? Can a green horse on the very first day of training be taught so solid to "stand statue still when being mounted" that you can grab mane and swing up on it bareback from the near side, then swing off far side, and swing back up from far side - repetitively - with the horse standing perfectly calmly and statue still like this is "nothing new" - like it is an old hand - when actually nobody ever mounted it nor did any prep work toward mounting it until that very day - probably no more than 45 minutes to 1-1/2 hours earlier. -- To answer the question at the beginning of this paragraph - yes it can really be done - by "starting right."
How? By intentionally defusing each of the 4 anti-predator triggers. And yes you can and "optimally" you should do it all in one session. Don't underestimate the horses ability to learn - prey animals are precocious learners - they learn MUCH quicker and remember longer than predators (including humans) because their very lives can depend upon it. They aren't going to think like a human (predator) but they are the very best thing on earth at thinking like a horse (prey) - so your job is to teach that horse that even though you are a predator you aren't going to eat it - to teach it that when you hop on its back there is no reason to be afraid and thus there is absolutely no reason to try and get rid of you by bucking you off.
If you have a good high round pen you can do this with the horse at liberty - but I know that lots of people that want to start a horse under saddle don't have a good high round pen. So I am going to outline how to "Start Right" assuming you do not have a round pen but you do have a halter broke horse and a 30-35 foot rope. The rope can be put on in the form of a "come along" type rope halter or just clipped or tied to a well fitted halter. The rope is to allow you to keep the horse workably close even if you goof up as you learn to judge the difference between "glued to you" attentiveness and "that's too scary - I'm out of here" - which is key to making the first part of this work.
So with the 30-35 foot rope attached to your pupil's halter you step back away from it so that you are basically 25-30 feet away and directly facing its head (horse facing directly toward you - looking at you intently). If you have done well with your halter breaking earlier the horse likely will want to follow you everywhere so getting it to stay put while you move 25-30 feet away will be the first thing you have to accomplish. Sometimes stamping a foot is all it takes or stamping a foot as you take a large step toward it - both of which are "horse talk." Other times you may need to partially or fully raise hands over head. The idea is to do the least possible to get it to stay put while you put that 25-30 feet between you and the horse pupil by moving yourself 30 feet away. You don't want them to skitter away scared - that would indicate you over did it - you just want them to stop following you and stay put - ON YOUR COMMAND - with their attention glued on what you are doing. (And that correctly implies that they should also follow you even halterless - ON YOUR COMMAND.)
Once you have the horse standing about 25-30 feet away from you and facing you start lifting your hands up over your head - watching the horse INTENTLY while you do so. The rope should initially be held up off the ground a bit between you and the horse to help lessen odds of stepping over the rope or getting tangle-spooked - until you have got the horse standing "steady still" facing you - very intently watching your every move. At that point I favor keeping the rope in one hand but laying loose on the ground - not taut between me and horse - so as to eliminate movement transmission via the rope to the halter because of your hands moving up and down rapidly as they soon will be. You are starting to defuse the "rapid nearby upward motion" trigger. Putting your hands up over your head then straight back down by your head then rapidly over your head again is done to simulate you jumping up and down - without you wearing your legs out prematurely (because you are going to need them well rested later on.) If the horse is super skittish then move hands up and down slower. If the horse is NOT totally paying attention then (for that horse) you need to speed up how fast you move your hands up and down, see if that is enough, if the horse is still not riveted on you then move closer while you keep doing rapid up and down hand movement - beside your head to full upward arm extension - repeatedly. Now the key is get closer and closer all the time maintaining the correct demeanor in the horse pupil - namely - ears pricked forward, eyes GLUED on you - VERY intent but not leaving - keep it right on that edge of thinking about leaving but NOT even slightly starting to act on it. You keep moving forward as the horse relaxes - gathering up rope as you go - all the while doing the up - down hand motion to simulate jumping. If you goof up and the horse starts to leave then get it turned back around facing you, back off, start over - you keep trying and you will learn how to read the horse and keep that dynamic tension just right. Sooner than you can imagine you will literally be up with the horses nose on your chest or belly as you rapidly put your hands to full upward arm extension and back down by your face over and over and over and that horse just stands there like "this is just a normal everyday happening" - "nothing to fret over" - certainly nothing to be afraid of. Good job, good start.
Now you move over to the side of the horse about 20-30 foot back with you facing directly toward the horse's withers or just behind the withers. And you do the whole rapid full upward arm extension approach all over - with horse standing statue like as you approach from the side. Odds are you will be up by its side with your belly pressing against its barrel MUCH quicker than you got up to its head. And as you extend your hands to full arm extension with your belly pressed against the horses barrel you are now somewhat working on defusing a second trigger - the "something on back higher than head" trigger. Once you have this done thoroughly - meaning you can do super rapid upward arm extensions repetitively with your belly pressing against the horses barrel - then you go over to the other side and repeat it over there.
Now you have largely defused the "rapid nearby upward motion" trigger from the front and both sides - now do the same from directly behind the horse BUT do NOT approach within kicking range - stop outside kicking range - about one and a half horse lengths behind the horse and do NOT approach any closer.
If you read the horse well then the horse should not have moved at all as you did all this work defusing this trigger from front, sides, and rear. Seriously - the horse pupil should stand perfectly still the whole time.
Now walk up to the near side (left side) of your horse facing the riders groove (the area of the back where the rider naturally sits on a bareback horse). Set your hands gently atop the horses back - don't press down with your hands but just start raising up on your toes then back down and work up to jumping up and down with your hands just resting on the horses back. You are now completing the defusing of the "rapid nearby upward motion" trigger and simultaneously starting to defuse the "sudden pounce on back" trigger and again working on the "something on back higher than head" trigger. Your hands are transferring a bit of your jumping motion even though you are trying to "just rest them" on the back. Once the horse gets that "ho-hum, boring" look then start jumping higher and start pushing down with your hands eventually holding yourself up with your hands on horse's back, your arms full extended, elbows locked. Do the all the way up to elbows locked deal a few times on the near side - then go over on the off side (right side) and start over with hands on back and build up to the same all the way up to elbows locked several times. The horse should NOT move as you do this.
Now you are well on your way to having 3 out of 4 triggers defused. On to the fourth.
Stand on either near or off side - your choice - reach over horses back and squeeze its barrel a bit - repeat this over and over building up to really squeezing for all you are worth - like you are hanging on for dear life. The horse should NOT move as you do this. Move back and forth a bit squeezing different spots on the rib cage. You are working on defusing the "squeezing pressure around barrel" trigger. This is important because you will grab on with your legs - so it is important to thoroughly defuse this trigger. Now go over to the other side and repeat. Yes it is important to do each step from both sides.
Now you work on getting the horse ready for things that happen incidentally when it is being mounted - like getting bumped on the croup with a foot if rider goes off balance OR like a leg dragging over the back when a person mounts bareback from a laying across back position. You do this by simulating those actions with your arm and hand as you stand right tight against the horse by the riders groove. Reach back and move your arm up into the hip like you bumped it mounting bareback then sweep it across the back and croup like you are dragging your leg across - once it is going "ho-hum, boring" about this then you simulate dropping your foot on the croup trick that newbies do or that any of us can do if we get off balance when mounting. Bump a few times until the horse is giving you that "ho-hum, boring" look.
You are now ready to mount up. First time up you go right across the withers - belly down across withers with more weight on the side where your feet are just in case the horse moves (which it should NOT do.) Basically even when you have a horse to a point of high trust-ability you still play it safe and plan for the worst - it is still a horse - and if it gets spooked by something outside your area and moves you want to land feet first not head first, right? Okay you will very rapidly find it is very uncomfortable laying basically with the end of your ribcage hooked on the withers. But the horse is least likely to have issues with you getting on the withers first - so do it that way even though it is uncomfortable. Get down, repeat several times - then go over to the other side and repeat. The horse should stand still the whole time.
Now once you have down the belly down withers mount repeatedly from both near and off sides - go back to the near side and mount belly down on withers one more time, then slide back into the riders groove. This will be a LOT more comfortable for you. Now use your arm that is toward the tail and do the arm sweep across the area you would brush your leg if you were mounting from this position. Get off, then hop back up belly down on riders groove and do the arm brush again. Repeat this a half dozen to a dozen times on the near side then go start at beginning of this step and do the process on the off side. The horse should stand still the whole time.
Go back to near side, belly down mount onto riders groove, swing leg up (but not over) and now actually sweep your leg back and forth - it is heavier than your arm. Now keep your head down but swing your leg over the rest of the way to riding position with your legs just hanging loose - not gripping. Now sit up. You are now fully mounted on the horse. Hop off and repeat several times from near side. Now go over and do it all from the off side just the same way. Each time you mount up fully do it just a wee titch faster. You should be able to jump up belly down, instantly sweep leg over to other side and sit right up. Now dismount opposite side you mounted from - mount from that side - dismount from other side - do this a half dozen to a dozen times. Horse should stand still the whole time.
Okay now you test to be sure you FULLY defused all four triggers by tripping all four in rapid succession. You do this by grabbing mane and swinging up "Indian style" onto your bareback horse pupil. Horse should stand still for this like it has been mounted this way hundreds of times even the very first time you do it - no kidding. Dismount opposite side and grab mane and swing up from that side right away. Do a half dozen to a dozen of these grab mane and swing up mounts, dismount opposite side, swing back up right away. Horse should stand still the whole time.
End the session with plenty of praise given your horse pupil.
Next session - whether the next day or two later or whenever. Repeat this session - start up closer - move up quicker - you should be on the horse within 5 minutes in the second session IF you did the first session thoroughly. And yes the horse should stand still the whole time. That second session is just a "drive the point home" session along with applying wisdom in assessing that the pupil retained what it learned the previous session. If somebody else has done the prep work then not actually mounted the horse I test the thoroughness of their prep work via doing this whole session starting closer (sometimes starting with belly against horse face or barrel) and doing the whole thing MUCH quicker - as a safety assessment practice. If the horse passes muster then I proceed from where they left off. --- Not a bad idea to "drive the point home" two or three sessions after your initial mounting session.
I also suggest doing the first several days to two weeks worth of training bareback. This will encourage you to work slow and easy rather than rush your horse pupil. Remember that it has to get used to lugging you around - so work it at a "walk and trot only" for a while while you teach it to turn, stop, back, etc.. All that kindergarten stuff can be done bareback. And introducing the saddle is another topic - for another time.