Belfast is easily divided into walkable quarters: The City Center spreads out from around the impressive, domed City Hall building and bustling Donegall Square. This is the best place for shopping, particularly along Donegall Place, which extends north from the square, onto Royal Avenue. Bedford Street, which travels south from the Donegall Square, becomes Dublin Road, which, in turn, leads south to the University Quarter, the leafy area around Queen's University. This is where you'll find the Botanic Gardens, art galleries, and museums, as well as a buzzing nightlife scene. Heading north from Donegall Place, it's a short distance to the Cathedral Quarter, which surrounds Donegall Street, and holds, as the name implies, the city's most important cathedrals, as well as many vast Victorian warehouses. Finally there's the Golden Mile -- the area around Great Victoria Street beyond Bradbury Place. It's considered the city's best address for restaurants and pubs, although it's a bit hyperbolically named. As locals say, "It's not a mile and it's not golden. But it's nice enough."
It's easiest to start in the center and then branch out from there. Perhaps tour the City Hall, then spend some time shopping for linen and china, then a walk to the Cathedral Quarter to take in the architecture, and finally dinner and drinks on the Golden Mile.
The sectarian areas, with the famous IRA and Protestant murals, are just to the west of the city center. The most famous of these are on the Shankill and Falls roads. It's perfectly safe to drive the roads and take photos yourself (locals are quite proud of the murals), or you could take one of the Black Cab tours if you want a guide to explain what it all means.