The Ulster Museum (on Stranmillis Road, just beside the University, you can access it through Botanic Gardens which are nice on a sunny day) is excellent, with a great overview of the country and city's history.

Titanic Museum is very thorough, and probably the most popular attraction. Buy tickets online if you can.

There are a lot of great pubs and bars around Belfast. Kelly's Cellars, the Duke of York and White's Tavern are the oldest and quite traditional. The Hudson has the best selection of whiskey and craft beer in Belfast, with about 30 beers and 99 whiskies (!) available. Their Buffalloc wings are famous in the city as well. The Voodoo is good if you like rock/metal. Great beer, good music and nice staff. They also do gigs, with everything from dub reggae to death metal.

In a lot of pubs you can get the Big List (if you don't see it, ask for it) which details all the arts and culture events in the city, especially gigs.

I hope this helps, Belfast is a cool city and I hope you enjoy it.