For a truly American visit, you just have to catch a ballgame. We are spoiled in Toronto since 1989 with having a quite revolutionary place for playing baseball. There's a huge monitor (used to be the biggest) and all sorts of bells and whistles, but there is something missing with the intimacy that you get at an American ballpark like PNC Park in Pittsburgh. Spontaneously we ended up catching a game while wandering around the city on our first night. The usual "I wonder where this goes" led to a great ballgame between the Pirates and the LA Dodgers. The view alone from our seats were worth the visit.
Of course there is also some great architecture and museums in the city to take in as well including the Warhol Museum and the Carnegie Museum of Art. They were a great escape on a hot day. The Phipps Conservatory was worth taking in in the heat, though.