I can really recommend Tara Lodge Hotel - small, nice,independent and great traditional breakfast.The surrounding houses look a bit rough on first site but no real problems and wonderful friendly and helpful staff. Easy walking distance to the Golden Mile (of bars and restaurants) and pretty close to the centre. Tends to be pretty highly rated on tripadvisor.com.
We normally see relatives so not that hot on sightseeing but a trip to the Giant's Causeway and the distillery there may be fun. I have heard that the Black Taxi cab tours of the different areas of Belfast seeing the murals and the history is very interesting.
Loads of bars and restaurants - really depends what they want but for traditional try The Crown Liquor,Robinsons and Lavery's. Couple of decent restaurants near the hotel too - Darceys and Benedicts (also a pub) (cheap early bird meals and a great Sunday lunch/carvery complete with a live band )
There are lots more upmarket places as well - Paul Rankin's Cayenne is nice and the Apartment Bar right by the City Hall - for cocktails and tapas snacks.
Don't know about live football but for TV football and a cheap and cheerful studenty pub with decent grub The Botanic Inn (The Bot) further up the road to the University is good.
I am sure more local people or those with more knowledge will give you more info but that's my recommendations