I was wondering what the basic difference in training is between a horse that is ridden, one that pulls a wagon, and one that plows is? Is there a difference in training or is it all basically the same?

I think it's simply a difference in training. Horses that are trained to be ridden are more smooth in their gait, whereas horses trained for carriage and plow are more rough and bumpy. Usually though, horses are trained in both aspects, pulling and riding.
A side thought. Plow horses are usually larger and more muscular and riding horses are usually the more finer boned type.