There is a ton to offer in Pittsburgh. If you want a sense of the history of the city, the Heinz History Center and Sports Museum and Carrie Furnace tour would be two of my must dos.
Stadium tours are probably a cool idea. The Steelers just opened up a Hall of Honor Museum at the stadium.
Strip District - Primanti's, Pamela's, Kelly O's, Deluca's (breakfast spots), 'Burgh and Bites food tour. You'll be able to find Pittsburgh merch here as well. The Heinz History Center is an awesome local museum with a sports museum inside as well.
Market Square - a bunch of restaurants in downtown. Millie's Icrecream and Milkshake Factory are nearby.
Northside - A lot of museum options. Carnegie Science Center and the National Aviary. If you're looking for something a little different, the Andy Warhol Museum, Mattress Factory, or Randyland are good spots to hit.
Station Square - shopping and restaurants, Gateway Clipper (cruise the river), the Inclines to Mt. Washington
Oakland - University of Pittsburgh and Carnegie Mellon University. Cathedral of Learning, Carnegie Museum of Natural History, Carnegie Museum of Art, Phipp's Conservatory, Schenely Park
Elsewhere - Pittsburgh Zoo and Aquarium. I'd also HIGHLY recommend the Carrie Furnace tour. It's a tour of an old steel mill and really highlights the history of Pittsburgh.
Into the outdoors? Drive about 1 hour 15 minutes to Frank Lloyd Wright's Falling Water. Down the road from there is the state park Ohiopyle with a ton of great hiking trails. Jonathan Run Trail (and some offshoots) is a great waterfall hike. Ferncliff Trail is an easy hike and Meadow Run Trail goes by natural water slides and a couple of other cool sites. A longer hike is Baughman Trail, which goes to a nice overlook.
There are probably a ton of good burger places, but I like Moonlit Burgers in Dormont. They're smash burgers and are pretty good.
