Belfast is a beautiful city - I recommend the black taxi tours if you are interested in the conflict and peace process. of course there is the Titanic museum, shopping downtown, St Georges Market, Crumlin Gaol and more. If you want to throw them all together there's the hop on hop off tours http://belfastcitysightseeing.com/hop-on-hop-off-bus-stops/they go to some of the same places as the black taxi tours, but if memory serves me correctly, the black taxi tour guides are less touristy, more lived through it... though this could have changed since I took the tour.
I highly highly recommend doing a day trip out of Belfast to the Giant's causeway - this is an amazing rock formation thing on the coast. Words can't describe - google it and go there