Albanach, Andy has mentioned a lot of he stuff that I would suggest, however there are some others that would be worth considering.
The Crumlin Road Gaol is fascinating if a little macabre. If the weather is good I would consider getting the train to Cultra (takes about 15 mins) and visiting the Ulster Folk and Transport Museum. It has two areas obviously the transport side and the folk side is outdoors and features a number of reconstructed buildings from mostly the 1800’s. The buildings are all staffed by actors and they give a wonderful insight into life in rural Ulster at this time.
If the weather is poor I’d go to the Ulster Museum near Queens University in the south of the city. 20 min walk from the city centre. You can go to botanic gardens up here as well.
There is an Andy Warhol exhibition on in The Mac until the end of Apr.
As for pubs the Crown is a must and I know you may like folk music from other threads so Fibber Magees next door to the Crown would be best for some live music on the sat.
A walk up Cave Hill if the weather is good gives great views of the city and further afield and if you want we could hook you up with bikes and a quick guide round!
Some links to the stuff I’ve mentioned
http://www.gotobelfast.com/things-to-do
http://www.robinsonsbar.co.uk/fibber.html
http://www.redbull.co.uk/cs/Satellite/en_UK/Video/See-Gee-Atherton-Chasing-the-Pack-at-Red-Bull-021243290358389
http://www.nmni.com/uftm/Collections/buildings/Rural-Buildings
http://themaclive.com/
If you are going to the Titanic centre I’d recommend booking tickets now as most weekends sell out. If you have any questions or want to email me feel free. Email in profile