I couldn't tell you what *most* trainers do, but I can tell you that most do more damage than good. Before buying a young horse, look into finding a trainer less than three hours from your home. If you find one you think you like, ask for the names of owners whose horses this trainer has broken- see if you can set up an appointment to talk with at least two of said owners, possibly even ride their horses.
I can tell you what we do. We first train horses in a roundpen in the indoor arena (but no, we don't "round-pen" our horses. We just like riding in them), and after the horse is confident and well-trained to the aids to stop, stand, go, walk, trot and canter (well, we train gaited horses, so we don't trot or canter- but I'll just adjust to suit the average horse and rider), back up, turn, etc... often a horse will have at least one little (or huge) problem during their training- whether it's a "go-forward problem", they're balky, spooky, a bolter, a bucker, a rearer, a nervous wreck, etc... we work through these issues in the round pen before we expect the colt to handle the issue out in the open. We also try to ensure that our horses don't drag their feet, move squarely and balanced, and with impulsion. Lightness is also a big factor- no one likes a horse that's hard mouthed and difficult to control.
We put a huge emphasis on being certain that a horse is relaxed and fully comfortable before moving on.
Once the horse is good in the round pen, we move on to the whole indoor arena, and after they're good in there, we have a riding track that borders our property, which is a controlled environment to practice riding outside in a trail atmosphere. I prefer it to an outdoor arena simply because the scenery changes, and the horse is always encountering new things. In an arena, they see the same things and learn to ignore them, but when they're out on their own and things change, they don't know how to cope. This way, they ride past a lot of different animals and vehicles and sceneries.
After spending time with the owner, we figure out what exactly this horse is likely to encounter in their new life, and prepare them for it. This means that if the owner plans on a lot of trail riding, we prepare them for trail work- we introduce them to a lot of different things they're likely to meet on the trail, and also make certain that they don't become buddy sour or barn sour, or bolty.
The main idea is to give the horse the tools to be ridden well and to behave nicely, to prepare them for whatever work is ahead of them, and to give them enough experience as to make a good match with whomever their rider will be.
