When you think "dressage horse" do you automatically think "warmblood"? There are plenty of good reasons to associate European Warmbloods with the sport, but there are other breeds, including ponies, often smaller and therefore easier for small adults to ride (and sometimes more affordable as well!), that have plenty of potential for the dressage ring. If you start with a horse you can ride that has three good, honest gaits, then you can develop it.
Morgans: As a rule, Morgans have a good temperament for dressage. They're very trainable and I have known members of this breed who competed all the way to Grand Prix. Some Morgans have a rather flat canter that limits their ability to collect or to execute big expressive changes. Or they may have a round stride without the scope of the shoulder they need to carry it forward. But they're often very good with the piaffe/passage tour and can make up some points there that they lose with the quality of other movements.
Connemaras: As a breed, Connemaras are extremely rideable. Both of my very successful Grand Prix "little horses," Seldom Seen and Last Scene, were out of Connemara mares by Thoroughbred stallions. The Thoroughbred is a good cross with the colder type of Connemara pony because it gives them additional size and a little more blood, but there is also a hotter Connemara strain that doesn't need extra blood! My two Connemara crosses didn't have spectacular movement when they started dressage but their movement improved as I trained them up through the levels.
Arabians: Collection is difficult for a lot of Arabians because they are bred to have a flat croup, which is rewarded in Arabian breed shows--while for dressage we want the opposite type of conformation, because horses with a flat croup don't naturally step under. Even if they're not actually built downhill, this trait causes them to go rather downhill. Arabs are nicely sensitive--I think that you should have a dressage horse that's as hot and responsive as you can manage--although their alertness to their surroundings can make it difficult for an amateur to get them on the bit and keep them there. Arabians tend to have straighter-legged motion in their gaits than we look for in dressage, not using their knees and hocks the way we need them to. In piaffe and passage, they may get the suspension but not the roundness.
Quarter Horses: The traditional Quarter Horse, bred to be a comfortable ride for hours of herding cattle around the countryside, is very quiet with a small easy-to-ride stride and doesn't use a lot of energy while working. This breed can be a superstar at the lower levels of dressage, though its naturally downhill build (great if you want to rope cattle) becomes a disadvantage for training to First Level or beyond.