If you have the budget and want to avoid more than 1 trip, the easiest way to do this is to find an agent in Europe (Germany, Holland, Denmark, or Spain usually) who will pre-select a bunch of horses for you to try. Then you take 3-5 days to fly there, ride all of them, and pick one. You can also go to Florida during winter show season and try a lot at once. I would never buy a horse without trying them if your intent is to find a good riding partner (vs rescuing, wanting a pet, breeding, etc.). There are agents who work domestically, but I don't know anything about them and you'll still have to travel to try the horses they find.
I agree that concentrating on finding a good trainer first is the best idea. Someone who knows you and your riding will be invaluable for helping you find a match, regardless of where it comes from. What you think you want and what kind of horse is best for you as a rider may not be the same thing and it's always good to have someone around who can guide you. A well-informed trainer can also give you a much better sense of the market and what people actually pay for horses vs. what people ask for.
What are you looking for exactly?