It’s seldom a problem to move a dog to the US, assuming s/he has been vaccinated. And the cost is minimal, just the cost of an extra bag. Realistically, though, you’re not going to be able to have your dog with you for at least one or two years while living on-campus if you’re at a 4-year school (particularly one that gives scholarship money).
Also, if you go somewhere where a lot of students live off campus later on, it creates an added challenge to find a place where the landlord allows dogs and your roommates are fine with it. Not impossible, just harder. You’d also have to think about whether your dog would be happy in an apartment, alone much of the day, if that hasn’t been his/her life so far. Either way you probably want to get your pup used to your being away for a while and being mainly with family members if you’re thinking of studying far from home.
Rules on importing a dog. I don’t know if there are any special conditions when the owner is on a student visa, but here are the basics: