I would recommend staying on-site. Doing so, for lack of a better way of describing it, throws you right into meeting people, and it makes the potentially lonely situation a lot easier to handle at first...plus, you end meeting a TON of people that you will hang out with during the entire program.
Trips home: it pretty much depends where you work. If you can get the time off, take a trip home and visit the family. I asked for a Fri, Sat, Sun and got them...traveled home to visit the family before Christmas. Just make sure to ask WELL in advance.