Health problems as in like ones they have recieved because of breeding, etc...is different between breeds, and their bloodlines.
Health problems in general include colic, founder, and laminitis. Horse colic - Wikipedia, the free encyclopedia Founder and Laminitis in Horses
As for choosing a vet, I'd ask around and see who people recommend. I asked a friend of mine whom she used, and I actually knew the lady who was the vet (I used to ride at her stables) so we "hired" her of sorts hehe and now she's my horse's vet.
For charges...it 100% depends on your vet. I cannot give you an exact ammount.
Spring Shots: approx. $180
Fall Shots: approx $115
Colic: approx $200-$10000 (depending on how severe)
Again prices depend on the vet you have