For night life, you're more or less looking at Armory Square downtown. There are a bunch of bars and a few clubs in that area and almost all of the places are open until around 2:00 am on Friday and Saturday nights. There are a few good bars scattered around other places in the city (Faegans and Colemans are two that I've been to), but I'd say the most activity is probably in Armory Square.
For other stuff to do, there is always SU sports as Mosh said. Football and basketball are the main draws, but the lacrosse team is usually one of the best in the nation and their games are pretty fun too. Syracuse has a AAA baseball team called the Skychiefs.....tickets are usually dirt cheap and it's a pretty fun way to spend 2 or 3 hours in the summer. If you're looking for a day trip type of thing, there are a few nearby towns that have stuff to offer. Skaneateles has a nice lake and is a nice place to spend a few hours walking around in the summer. If you like the outdoors, Ithaca isn't all that far away and has a fair amount stuff to do. You can also go to the track at Watkins Glen to watch one of the NASCAR road races (Bud at the Glen) at some point during the year if you're into that. Turning Stone Casino is in Verona, NY (~45 min away).
As far as apartments go, if you're looking to live around the University area, a fairly good rule of thumb is to look on the campus side of I-81. A friend of mine had a decent apartment in a decent neighborhood on the east side of the city and the west side around the Tipperary Hill area isn't too bad either. Otherwise, maybe take a look at some of the suburbs like Liverpool, Jamesville, or DeWitt.
One place you might want to check out is Syracuse.com. It's run by the local newspaper (Post-Standard) so you might be able to get some good information about nightlife and other stuff to do from there. That's all I got for now.
-Tom
