I would always be open to suggestions about what to do on my day in Pittsburgh, though I'm considering riding the T to the endpoints, and also riding the inclines (which I've done before.) I've actually spent a couple of tourist weekends in Pittsburgh, though the last time I did it was 15-20 years ago, so there might be some new things to see.
Yinz asked the right forum. None of these requires renting a car. My list:
Within walking distance of downtown:
Monongahela Incline (
http://www.portauthority.org/paac/SchedulesMaps/Inclines.aspx) —This is a must.
The Point (
Point State Park - Wikipedia) —Where the Allegheny and Monongahela meet to flow into the Ohio. Strategic prize for the French (Ft. Duquesne) then the British (Ft. Pitt). Historic blockhouse, and an excellent small museum.
Andy Warhol Museum (
www.warhol.org )—Just across the 7th St. (Warhol) bridge. A native son who left provincial Pittsburgh for NYC. Fascinating museum, quirky gift shop.
Mattress Factory (
www.mattress.org ) —An avant-garde museum focusing on large-scale modern “installation” art. Surprisingly fun. Bit of a hike but doable.
National Aviary (
www.aviary.org ) —A unique and wonderful exhibit of birds, from penguins to flamingos to eagles, many of them in approximations of their natural habitat. Surrounding neighborhood is called the Mexican War Streets, a collection of restored 19th-century rowhouses (they actually have nothing to do with the Mexican War), a gem of a neighborhood.
Heinz Regional Museum (
Home ) — Excellent museum, located in a converted icehouse. Its main focus is on regional history, from French and Indian war times to the present. Try your hand as a pickle-packer. And the restored trolley brings back lots of memories. This is on the western (downtown) edge of the Strip District, so you could combine it with an outing to the Strip.
Strip District (http://www.popcitymedia.com/features/updated2guideStrip102611.aspx ) —This neighborhood is along the narrow “strip” of land between the bluff and the Allegheny River, hence the name. It starts at about 11th Street, but its heart is Penn Ave. between 16th and 26th. Historically it was a railway terminus and headquarters of the wholesale produce business. It remains the food-lovers’ paradise of Pittsburgh: lots of very cheap and very authentic ethnic groceries, street food, Steelers crap, etc. Gritty, not pretty.
PNC Park (
https://www.mlb.com/pirates/ballpark) — The best ballpark in the major leagues. Yeah, the team is weak, but hope springs eternal.
Easy bus ride from downtown:
Pitt Nationality Rooms (
http://www.pitt.edu/~natrooms/) –These are classrooms, in Pitt’s Cathedral of Learning, each of which was adopted and decorated authentically (not just items and artifacts, but architecture) by a local ethnic group. They are beautiful and moving. They function as working classrooms so some may be off-limits on a weekday.
Carnegie Museums (
http://www.carnegiemuseums.org/; the art and natural history museums are in Oakland, near Pitt). Wonderful place, with one of the best dinosaur collections anywhere and good permanent and rotating exhibits.
Phipps Conservatory (
http://phipps.conservatory.org/) — Fully the equal of the U.S. Botanic Gardens. Within walking distance of the Cathedral of Learning and the Carnegie Museum and Library.
Frick Art & Historical Center (
http://www.thefrickpittsburgh.org/index.php ) — Wonderful small museum, mansion (don’t skip this), café.