Primanti brothers is an excellent and iconic sandwich place in Pittsburgh. Wikipedia link has a list of notable mentions this enterprise has received. There are multiple locations in the city.
The Cathedral of Learning is the second tallest university building in the world and houses 29 nationality rooms which are a must visit.
The Duquesne incline is an incline plan railroad first started in 1887, scaling Mt.Washington and having a brilliant view of the Pittsburgh downtown and the confluence of the Allegheny, Ohio and the Monogahela rivers.
If you are the type, there are a bunch of museums including The Carnegie Museum of Natural History, the Hienz History Center detailing the sports legacies of the Steelers and The Andy Warhol Museum (Andy Warhol was a native of Pittsburgh and a leading figure in the pop art movement).
Also, if you are interested in bridges, Pittsburgh has the world record for most bridges, I beleive 446 of them, beating Venice which was historically considered the City of Bridges.
P.S. This does not strictly come under the city of Pittsburgh, but an hour's drive from the city is Frank Lloyd Wright's famous Fallingwater.