Classic Pittsburgh Attractions:
The Carnegie Museums Natural History has a fantastic T-Rex skeleton. Parts of Slience of the Lambs were filmed there, as well as at the Soldiers and Sailors Memorial Hall and Museum. Warning: Parking in Oakland can be kind of a pain. If you do get that way though, make sure to also check out the University of Pittsburgh's Cathedral of Learning, especially the Nationality Rooms.
"The O", Pittsburgh's most famous hot dog stand. If you order anything larger than a small order of fries, make sure you have friends to help.
Dave and Andy's Homemade Ice Cream. If I remember correctly, their waffle cones have a solid chocolate plug at the bottom to keep them from leaking.
Everyone raves about Primanti's but honestly as far as the food itself goes I've never been impressed. Yeah, I said it.
Finally, some of my personal favorite restaurants, although to be fair I haven't lived in Pittsburgh since 2008 or so, so your mileage may vary:
Point Brugge neighborhood Belgian bistro
Aiello's Pizza. One of the top pizza places in town, next to their neighbor and rival Mineo's
Last, but the opposite of least: Pamela's. Best pancakes (seriously, THE BEST pancakes) and greasy, delicious diner breakfast anywhere on the planet. Your tastebuds will not regret going here...your waistline might.