You probably want a mountain bike if you are going to ride a lot on rough, rocky trails. It can certainly ride on roads too, while a road bike doesn't do as well offroad. Cyclocross bikes are basically road bikes that can handle some off road riding but not as well as a mountain bike. You certainly don't need to spend anwhere near what you are suggesting on your first bike - $500-$1000 will get you a perfectly good bike and even less is necessary for a good second hand bike. You can browse this site or
www.sheldonbrown.com for lots of information.
How old are you, what do you weigh, what seasons do you expect to ride in?