Hate to be a bit stereotypical but "our wee city" is brilliant for a few scoops!
Top tip would be to do the Black Taxi Tour! Takes you around the city and although its a bleak history, the trip is very important!
Do take in some of the bars... Laveries (to catch the match), then some of the more interesting bars, The Crown, Fibbers, The Empire, The Duke Of York, The Hatfield, Kellys Celler, or Whites Tavern. Some will have a bit of the old "diddly dee" playing too.
Foodwises (as someone who has eaten in about 75% of places in Belfast) id say - £40+ pp, Cayenne, £30+Beatrice Kennedy or Tedfords, £20+ Mollys Yard - £10+ Benedicts (if they do Beat The Clock at weekends then its amazing)
There is loads to do on the weekend, St Georges Market, Titanic Quarter, Botanic Gardens, and dont forget the pubs!