Pittsburgh is one of America's greatest cities, you'll be happy to visit. Fallingwater is about an hour and a half or so from downtown, I've been there in the past few months but can't remember exactly how long, but its in a fairly rural area so it takes a while to get there.
So far as the city goes, you'll want to do downtown, Mt Washington, go east to Oakland, Shadyside, Bloomfield, and even venture into East Liberty if you like old churches. There is a Presbyterian church that is over 300ft tall with ornate architecture and isn't a structure you find often in America.
If you are a museum addict, Pittsburgh has world class museums. Its the home of the Carnegie museums and the Natural History and Art museum is a cool place to visit.
If you are into the arts and theatre, make sure you book some tickets for the theatre district downtown. Pittsburgh has about 6 theatres within a few blocks in the north side of the Golden Triangle.
If you like art museums don't miss the Andy Warhol museum, its 8 floors of his work and is just across the river on the north side's riverbanks.
If you want hot nightlife, head to South Side. Carson St is one of the busiest club districts in the nation. You'll find a flavor to satisfy you, just search online for what kind of venue you prefer be it electronic, grunge, rock, etc. If Carson St is too college kid oriented for you, try Station Square just across downtown, or downtown's clubs. The Strip District has some nice venues as well, but they are more sketchy at night.
There are a few comedy clubs in the city, but the newest one is actually in the new Waterfront area which is a suburban/new urban mall type deal in Homestead, its only 20-30 mins from downtown depending on traffic.
I recommend getting a downtown hotel, the transit system here is top notch and can take you anywhere you need from there, and the downtown subway is free of charge within the 4 stops in the Golden Triangle: Gateway Center, Wood St, Steel Plaza, and 1st Ave Station.
Also, if you're in Oakland you'll want to take a self-induced tour up to the top of the Cathedral of Learning and tour the international study rooms in the base. Walk along Forbes and 5th avenues and you may soak in some nice architecture, such as the War Memorial complex where Obama had his speeches while in town.
Of course all the typical tourist stuff is available, the Gateway Clipper is in Station Square and they do 1 hr water tours for a fee. Search google if interested. There are also Ducky tours that do water boats that drive on land and are a seamless tour.
Enjoy!