If your DH loves history, there is so much here! Beyond just museums, there are plenty of historic sites, particularly during the Carnegie/Rockefeller era. Fort duquesne was a major battle site and outpost for the French and Indian War. If you can catch a show downtown, Heinz Hall is a stunning historic landmark, same as the Benedum. Another place of interest near Pittsburgh may be the Flight 93 Memorial (the flight that crashed, but would have been involved in 9/11).
I’m biased for the Museum of Natural History because my uncle is actually one of the taxidermists there so I grew up going behind the scenes. There is also the Carnegie Museum of Art as well close by. There is also the Heinz History Center if you want to check that out, and the Bayernhof Music Museum. And, of course, for the kid in you, the Carnegie Science Center!
I second Kennywood (it’s an amusement park, but also has an entire section devoted to the historically preserved rides which your DH may be interested in. There are some old racer wooden coasters, and my favorite, the Thunderbolt!)
Schenley park is nice for walking around, but I would say Frick Park (named after the general manager of Carnegie Steel, the same man who caused the Johnstown Flood) seems like it might be more to your liking. There is a museum there (the Frick Art and Historical Center), and it is beautiful to walk around during the day.
I second and third Phipps Conservatory, it’s beautiful. There is a butterfly room where the butterflies land all over you. People have mentioned the Zoo (I highly recommend PPG Aquarium), but the National Aviary is also here if you’re into birds.
If you are willing to travel a little further, and love architecture like me, you could go visit Frank Lloyd Wright’s FallingWater.
Pittsburgh is also home to one of the 4 sites in the country where Hofbrau beers are brewed in the United States for distribution (using the old German recipe), and there is a huge replica of the brewery in Germany, complete with great German food and a lovely outdoor beer garten. It’s in South Side Works.
The Strip District is a fun place to walk around and shop during the day, there are a lot of vegetable and fruit stands, a fish market, etc. At night, a lot of the hottest clubs are down there as well.
Like a lot of people have said, downtown is historically dead on weekends/after business hours, though there is a sudden influx of young people living downtown now. Market Square in downtown has a lot of hot restaurants, and there is a 3 restaurant stack (Sienna Mercato) near there which has a two restaurants and a rooftop beer garden; the bottom restaurant specializes in meatballs — you can pick your meat and pick your sauce and try different combinations.