Experience over a degree wins out with horse training - every time. If you are DEAD SET on going to college, I would go get a degree in something practical - business, accounting, whatever - as a backup, and work for a trainer part-time. Or just go work for a trainer.
William Woods has a nice program, but is definitely not an Ivy League school. (I live about an hour away from it.) I go to Truman, which does have an equine science program, but it's very small, which has its benefits and its downsides, but it's very affordable (for me, but it's a state school and I live in Missouri). If you are DEAD SET on going to college for some sort of equine degree I've heard good things about St. Andrews. I showed there once. They've got extremely nice facilities.