If you are inexperienced at dressage, even if you're quite experienced at other types of riding, just not dressage, it's still very difficult to choose a dressage trainer/instructor.
To some extent it depends on your goals. Are you eager to ride in the Olympics, national or regional, or even local recognized competitions? The standards for how well one needs to ride in those, is increasing every day. Then you've got to be awfully worried about exactly what this person is going to teach you. While most incorrect training can be redone (at rather great expense and time!), some cannot, and becomes entrenched.
Do you just want to have some fun, do some training level now and again? Then you may be a lot more concerned about how near the trainer is, what he costs, and how nice he is to you. Someone who really pushes you and is demanding, you may not like.
Are you going to dressage for a certain reason? To get control of a difficult horse? To learn the advanced movements? Knowing that helps you choose.
Generally, the person who wants to compete, will put up with a lot more ah...'personality', and cost is not as important, nor is location. He's going to make a lot of sacrifices and work very hard, to get where he wants to go. He wants someone who will push and demand, and even, be very critical of him, even hard on him.
Experts aren't always 'cuddly' -experts in anything. People who really dedicate themselves to dressage have a deep passion about what they do - they LOVE what they do, doing it well is a PASSION to them. And they want others to be good at it, even great.
That isn't for everyone. Some people want to just have fun in dressage, get encouraged, enjoy the day, there's a trainer out there for them, too.
For someone who has specific goals, it's awfully difficult these days. There are a great many people 'hanging out a shingle' in dressage, who really should not be teaching. Trust me - ANYONE can pick up a lot of 'dressage words' and toss them around and sound great. But - many lack the nin depth experience of their own, of having taken a number of horses up the levels and of working under more experienced people. This is just not something you learn from books or videos, or a few short clinics.
Too, it's actually awfully difficult to teach dressage. It is. It just is. The student has to do so many things at once and the instructor has to somehow communicate to them. You're often dealing with horses with limitations, and people who have limitations too(tension, confusion, etc). What works with your horse, might not work with theirs. The rider may do exactly what you say, very correctly, and the horse still not do it. Teaching dressage, doing it well, it just isn't easy.
So please, if you can, if you see any instructor yelling, insisting, even demanding, don't worry. Urging, pushing, even demanding, that's not unusual in any kind of riding instruction. Often instructor and student have been together for years, and they may kid, tease or even scold in a way you might not be used to. Because it can be a lot of work, there is often a kind of very dry humor among riders and instructors. You may be seeing people who have known eachother for years and are very casual with each other.
I still don't recommend excusing really bad behavior. An instructor should not personally insult or frighten or overwhelm riders, and he should be able to 'adjust' to the casual student who wants fun, AND the more intense rider who is on a blood-and-guts mission.
An instructor should also enjoy all types of horses. He may say honestly that a horse with certain conformation or gaits may not do well at upper levels or score highly in competition, that's just honesty. But he should like and enjoy all horses.
Don't be swayed by people claiming they 'worked with' so and so the famous dressage expert. Most people have 'worked with' famous experts - but perhaps for only an hour at a clinic now and again, and without affecting overall how they teach or train. A long apprenticeship with a good trainer is good, and can lead to a person picking up a 'system' from the lowest to the highest levels, instead of a disconnected 'bag of tricks'.
Awards aren't always valuable. There are a lot of 'all breed' awards that are easy to get, and people can pile up an awful lot of ribbons by picking carefully which shows they go to, especially at the upper levels. It's called 'savvy', lol.
Even so, there are certain competitions that are very hard to do well at, and certain awards that are hard to earn. If you can, familiarizing yourself with that can help you 'read' who someone really is.
USDF instructor certification is also a good thing to see. It means the person got taught how to teach dressage, and did well enough in the classes to get a certificate. That can be very good, though many trainers/instructors are too busy to take time off work and go get the certification. It takes a big committment.
Have you got a friend who's been in dressage for a long time, done what you want to do in future, and might be able to recommend someone in your area? I mean, keep in mind that many people develop EXTREMELY strong ideas about THEIR trainer being Heaven on Earth and everyone else being the Devil. Not all of the recommendations you hear are going to be very fair or balanced. You have to kind of filter through at least some of what you hear.
You can go and watch of course, but you won't be able to tell if the horses are working correctly - to be honest, many people, even after years in dressage, can't tell. It is difficult - it's possible for horses to do flying lead changes, piaffe, passage, extended gaits, and look extremely impressive and exciting, yet still be working incorrectly.
Unless the rider is an absolute beginner, you should see active, energetic gaits on the horse - all gaits, walk, trot, canter. This should apply to most students as well as the trainer himself. A few students, of course, will be too timid to have their horses go energetically, but overall, you should see very active, forward horses.
Keep in mind that if you come from another riding style, even the very best work will look strange to you. The use of the reins and leg, especially, are very, very different.
If you're experienced with horses, you WILL be able to tell if the horses are clean, healthy, not lame, and appear to be fairly contented and happy. Good trainers generally have neat, tidy facilities, with good quality hay and feed, clean stalls (I mean really clean), ample bedding. You should see turn out paddocks for horses. All work and no play isn't how a good trainer's program goes.
If you watch people get lessons and watch the person ride/train, you'll be able to see if the solution to every problem appears to be draw reins or other appliances. A good dressage trainer does not rely on appliances like a gogue, Chambon, or draw reins.
You WILL see side reins when longeing, that's very normal. And while a good many people misuse side reins, their presence in and of itself isn't cause for concern.
A very good trainer will occasionally allow a very physically weak, overpowered rider to learn and put in the miles s/he needs to get fit, while using draw reins, especially when they're mismatched with a horse that is far too much for them and insist on not selling the horse.
But a good dressage trainer will in general, not make much use of appliances. He might use something for a day or two, but genereally, no, you should not be seeing a lot of extra gear on horses. This is going to be quite unlike other riding sports, but again, the goals are different so how you get there is also going to be different.
You should see simple snaffle (ring) bits, much like what's allowed in competition, with very few exceptions. Pelham bits are not allowed in dressage competition; if you see a lot of bits that are not allowed in competition, you may either be looking at students who refuse to change their bitting, or a trainer who is taking a lot of short cuts - trying to solve training issues by switching bits. Again, ok in other riding sports, not a problem. Here, a problem.
You will see drop nosebands or flash nosebands on lower level horses; again, their mere presence is not cause for concern.
You will see some double bridles (curb and snaffle reins, two bits) on the more advanced horses. Again, their mere presence, no cause for concern. If the horse is very young and green, or the double bridle is being used as a 'dodge' to avoid reschooling, again, a problem, but it would be harder for you to detect that sort of thing.
It isn't at all unusual for a very good dressage trainer to have horses that are very, very 'naughty', especially for his own competition horses. He may have a very different standard for the shaky beginner's horse where if the horse is silly, the poor rider may fall off.
The trainer may also be retraining horses that have very serious problems, so you may see some rather outrageous behavior, or he may believe, like many upper level trainers, that allowing a certain amount of exhuberance and naughtiness prevents horses from becoming sour and dull.
So again, the mere presence of some horses doing well, naughty things, not much cause for concern.
If you see excessive use of the whip and spur overall, blood on horses, anger, temper, inappropriately intimate behavior with students, poor standards for horse care - take my advice - get in your car and leave.