Breed really depends on a lot of things. Some breeds are known for being smaller in size, so it depends on how tall you are. Also what discipline you are doing, western (barrels, rodeo, etc) or english (jumping, cross country, dressage, etc). Most breeds will be able to accel at either, just depending on the horse itself. My friend has a very successful hunter/jumper Quarter Horse, while my other friend has a Thoroughbred who doesn't like jumping at all.
I have a Selle Francais who is very athletic and loves to jump. He is a great horse, sometimes feisty, but very trustworthy when being riden. I spent $1500 to buy my horse, as most of my friends also spent around the same amount for their horses. Keeping the horse is the expensive part.
Shoeing a horse depends on a lot of factors. Some breeds, like thoroughbreds, typically just have bad feet, while others may have great feet. It also depends on the location and weather. I see a lot of horses have to get shoes when it rains a lot because their feet get so wet and soft.
My horse is usually fine without shoes, but usually during rainy season he has them cause the water gets to him. The farrier has also trimmed him too short before causing him to need shoes.
Hope this info helps out! Good luck on finding your 'perfect' horse!