Whether or not you should do this has been discussed a few times already in this forum--this very thread, in fact. The short answer is, "It depends." Your qualifications, their school, the professor you want to work with.
As far as who you should contact, I would start with a short e-mail to the professor you want to work with. Introduce yourself, say why you're interested in their work, mention that you'd like to visit their campus, and ask if you can schedule a meeting with them while you're there.
To be sure, if you aren't invited, don't expect them to pay for anything.