What to Look for in a Trail Horse
Most people think trail horses do not need to be well trained, but in fact they need to be better trained than the arena horse to be safe. On the trail you are not in an enclosed space, you need to go over and under obstacles, thru water, over bridges, open gates, back up and turn in small areas, go up and down steep hills.  The horse must be surefooted, athletic and have some endurance if you plan on going for long rides.
Horses that have been used for lessons, dressage, jumping, western pleasure etc. do not always make a good trail horse, unless the previous owner actually took the horse on the trail.
Things to look for are calm temperament, comfortable, willingness, surefooted, some athletic ability, will go over obstacles, backs up, sidepasses, opens gates, has seen bikes and hikers, not spooked from dogs or cows, goes thru water, trailers well, will go out alone and with others, minimal spooking. An older horse will have more experience but sometimes they will also have problems- barn sour, herd bound,  buddy sour, jigs, rears, bucks, etc. If you get a young one under about 8 you will need help from a trainer at some point to keep you on track. It is unfair to expect the young horse just to figure everything out and take care of you as well, so if your unwilling to put in some effort, get the older horse and then sell him later when your ready to upgrade or want more of a project.
If you want a trail horse then you need to try it on the trail not just in the arena. I look at many horses and I am always surprised on how many people leave out important information and some just down right lie. The horse needs to be sound for how often you will be riding. Always get a vet check, so you know what your getting, most horses do not pass the vet check 100% especially if they are older, but you do not want to find out you just bought a lame horse and it is unable to do hills, go down hills or needs some special shoeing that cost 200 dollars every 6 weeks.
 Which breeds are best for trail? Quarter Horses, Gaited Horses and Arabians are the most common. So lets take a look at different breeds of horses.
 Gaited horses are wonderful, usually surefooted, good temperaments, smooth gaits and very willing. Remember I said usually, some are very pacey which is worse than a trot and some have been ridden so fast they don't have a slow gear. Some of the gaited horses will be calm but others are hotter depending on the breeding. Also if you buy one over the internet in tennessee or kentucky be aware that it probably won't back up well, sidepass, or do gates, as they seem to just ride them forward. They usually have not seen a bike or hikers because their trails are totally different then our busy trails, so you will have to put in some work getting them use to these things. Lastly some have not been taught to canter and teaching them is not for the weak in stomach as they pace and canter at the same time when they are started. The best thing about these horses, smooth, sweet, gentle, lots of endurance and cover lots of terrain in a short period of time. 
Quarter horses are some of the best trail horses. The Quarter Horse is one of the most versatile horses and they have amazing temperaments. They are sure footed, calm and willing. They usually will tolerate many things and many riders that hotter horses would not. They are usually easy to train and easy to get along with. Again that is not all of them, the performance horses- reiners, cow, rope etc. some of these have too many controls for the beginner rider and can be very quick to go forward or sideways and you better be sitting back when you say whoa, as they stop on a dime. Make sure if you can find out the breeding that it is a calm line not a hotter line of breeding. Best things about this breed, calm, willing, slow, sturdy. This is a great family horse, ranch horse or for someone who likes to mosey down the trail with their friends, you can ride them for long periods of time, but you will not keep up with the endurance of a gaited horse or an arab and that is why you will not see many in endurance races 
Arabs are very smart horses and also quick. They will figure things out before you do sometimes. They are surefooted and can go all day, but they tend to be more spooky and usually spin when spooked, if you don't have a good seat, you 'll be on the ground before you know it. Best things about this breed, endurance, comfortable, versatile horse, you will see it in many different equestrian sports. This is a great horse for someone who likes to move out at a trot, cover lots of terrain and likes to ride a lot, these horses are hard to wear out and that is why they excel at endurance. 
Draft and draft crosses are great horses. These horse usually do not care about much. They have calm temperaments and are good husband horses. They can be pushy and stubborn though if not handle right. Since they are large they are use to getting their way if they are allowed to. So make sure they have good ground manners and that they can be contained in a fence, some have learned to lean and just break the fences. Also due to their size single tracks may be difficult, they tend to be slow and getting them to canter can be difficult. They are otherwise gentle giants and usually great family horses as the kids can all ride at the same time. Best things about this breed, not spooky, sweet, gentle, willing, great for a family horse, someone who likes to walk on trail and rides on fire trails, you will have problems getting under low trees and opening gates on the larger ones. 
Morgans are great willing, calm, surefooted, usually a smaller breed like the arab. Now a days it seems hard to find a good one. Many now are bred hotter for the show ring, but if you find a good one they are great horses. Best things about this breed, sweet, gentle, willing, smart, comfortable, some are gaited, versatile breed. 
Mustangs are great horses but not for everyone. These horses still have a lot of natural instinct, so if you can't handle a domestic horse, then these horses are way above your level. They are usually best with one person, and they will do most anything for that person. They need to form a bond with the owner and trust can be a major issue with some. They will be cautious of things out of the ordinary, but otherwise are surefooted, tough and well made. Many can stay barefoot all their lives. There are different lines, so some will look more spanish, others drafty, if it was trained from a foal, you will have a better chance, an older horse that was rescued and broke can have some difficulties with a green owner. You really need to know what you are doing in other words, or stay with a good trainer for a couple years before going on your own. 
 Thoroughbreds are bred to run, they are great performance horses but the calm ones are hard to find although it is possible. They tend to be spookier but not as much as the arab, sometimes have poor feet and if previously run on track, may have old injuries that would prevent you from riding hard trails. Best things about this breed, they are fast, larger horse, versatile, can jump, best for someone who knows how to ride, has a good seat and balance, would do well in eventing and jumping or dressage. Hard to find calm ones for the trail but possible. 
Warmbloods are not bred for the trail either. This is another performance horse and although they are beautiful, they usually are large and due to this more difficult to maneuver on the trail. They can be quite calm but find out what they did previously, sometimes they have been pushed so hard in dressage that they have started to buck  or rear and everyone thinks well now he can just go be a trail horse although he hasn't been ridden on the trail. Best things about this breed, larger horse, athletic, good at jumping and dressage, would do better on open trails with room due to size and temperament. 
The friesians  I would consider with the drafts but know that they do not tolerate the heat as well as other horses. Have to be very careful on hot days, they walk slow on trail, like a quarter horse, larger ones may cause difficulty in opening shorter gates on the trail. Expensive for a trail horse.
The andalusians and spanish breeds are smart, willing and usually gentle. This is a hotter breed. Be careful of a deal on one of these, it means someone messed them up. Also if he piaffes, spanish walks, beware if you don't know how to do it, the horse has had this drilled in so much that they usually start to do it when they are confused. Some of them have also been taught to rear and do this when confused so you need specific directions from that owner if you decide to buy him. These horses are beautiful and expensive, if you find a cross like an azteca, then you might be able to afford one for the trail. 
These are just a few of the breeds out there, any horse can be a good trail horse just depends on what your looking for. Most importantly eveyone wants to save a buck, do not let this influence you on the horse, ah but he's so cheap. Occasionally you can find a great horse for cheap, but most the time there is some hidden problem waiting to pop up. A good trail horse that has the training will cost 5000 and up. This is one you should be able to just get on and go once you learn how to use the controls. If you spend less money you will end up putting in the money sooner or later anyway, either in training, maintenance or in hospital bills. No one ever believes it, until they end up wreck, on the ground, hanging on for dear life as the horse is bolting away. 
Be aware that although we all love pretty and flashy horses, do not let this be the major factor in your buying, it is better to get a different color than you wanted then to have the horse you always wanted but can't ride it or even control it on the ground.  I have seen this many times and it is just a sad situation. If you are still trying to figure out what breed, think of what you see at the trail rentals- Quarter Horses, Draft Crosses, Gaited Horses, and sometimes a Mustang. They are usually older, with lots of trail experience. They do not want people to fall off, it would be bad for business.