Go site seeing. It is a great place. If your into history and museums go to the Carnegie Natural History Museum. It is ranked within the top five of the country. I also imagane you'll be arriving via plane. If that's the case make sure you are paying attention when exiting the Fort Pitt Tunnels. Its a site to behold. Another fun thing to do is go walking down the strip district. I'll add more as I think of it. The West End Overlook is the best place to see the city and the rivers. You could also ride the inclines, there are two. They're a fun and almost scary way to get up to Mount Washington and back again.
Edit: What kinds of things are you wanting to do in America's Most Livable City (twice)?