First, you could visit the headquarters of Sinn Fein, which apparently has a gift shop (closed unfortunately on Sundays, when we went to scout it out).
In a similar vein, you can see the many, many interesting (and some slightly unsettling) political murals, which line many a wall in Belfast. The following two are on Falls Road (Catholic/Republican neighborhood). This one is from the "Solidarity Wall", which has murals of the Palestinian conflict, Fredericks Douglas, and Che Guevara.
From around Shankill Road, another Protestant/Loyalist neighbor. We saw this one while on a "Black Cab" tour, where a cabbie takes you around different parts of the city and talks about the conflict and The Troubles and all that jazz. The cabbie stopped and let us look around and see all a bunch of the murals in the area. When we came back, he asked "Did you see the gunman that was following you?" We immediately freaked out a little bit, but it turns out he meant the gunman on this mural. It's a bit creepy because it's like those paintings with the eyes that follow you - no matter what direction you're facing or where you're looking, the gun is indeed always pointed at you.
So, after you're done walking around segregated communities, listening to this city's (still) depressing history, well...you might need to go get some drinks. Lavery's is a good place for this.