Good bars and clubs to go to in the City centre are:
Skye
M Club
The Botanic Inn (personally I don't like this place but it is very popular so I'll recommend it anyway)
Stiff Kitten
Rain (Used to be called Milk, in Toome Street)
Yello
Apartment
The Northern Whig
Cafe Vaudeville
Ollies at the Merchant Hotel
There are also a number of bars, clubs and restauants in the Odyssey Complex (If you are staying in the city centre, you will need to get a taxi to the Odyssey)
For eating out, you can also get food at Cafe Vaudevlle and The Northern Whig, that I have mentioned above, and I would also recommend:
Taps - Tapas bar in Cathedral Quarter
Made in Belfast
AM:PM
La Villa Italia
Deanes at Qeens
The Morning Star -I've included a map of how to get to this place as it is down a little side entry, it isn't much to look at but the food is fantastic, especially if you like steaks:
http://maps.google.com/maps?hl=en&rls=co…
For shopping, make sure you go to Victoria Square, there are also quite a few restaurants there too:
http://www.victoriasquare.com/
I can't recommend a particular club that is good on New Years Eve, as I tend to avoid Belfast on New Years Eve, due to the fact that the taxis charge double fare and it is hard to get one, so try if possible to go to a club that is within walking distance of your accommodation to avoid getting ripped off in a taxi!