It depends what sort of things you're interested in - the Titanic Museum is good, as is the nearby Nomadic. You could probably fill most of a day with those two if you were interested. There's also HMS Caroline, which isn't too far from Titanic, and is also worth a visit.
The Ulster Museum is interesting, and it's within Botanic Gardens, which is nice for a walk around. It has a palm house and a tropical ravine which are worth calling in to (neither is very big). The gardens are beside Queens University, which you might like to see.
Crumlin Road Jail is worth a visit if you want some Troubles history.
The open top bus tour is good for getting your bearings and seeing lots of the sights (things like the peace walls, which are basically just big walls - I've never seen the point of going out of your way to see them, but you drive past on the bus tour and get the general idea).
If you're more into shopping, Victoria Square is your best bet, but the whole city centre is walkable and has some nice shops.
City Hall is quite interesting.
If I was spending a weekend, I'd probably do Titanic, Nomadic & HMS Caroline on the Saturday, and then on the Sunday id take the open top bus tour, and get off at Botanic Gardens to do the Ulster Museum and Queens; then in getting back to the city centre I'd do City Hall.
It's a small city, and fairly easy to get around. If you plan to stay near the city centre there are a few hotels which are good. There's a new one called Room 2 which has just opened and calls itself a "hometel" - you can cook/bring in your own food, or go to the bar to eat. It looks nice from the outside at least!