I have family in Carmel, so most of the time I've spent in Indianapolis as an adult has been in the 'burbs. But I went over there for a concert a couple years ago and stayed over and spent some time looking around in the city, and was pleasantly surprised.
Broad Ripple is cool, as noted.
I think the Art Museum there is very nice, although that may not be super fun for the kids. Haven't been to the Children's Museum since 1985, but I loved that place when I was young. I also liked the Indy 500 Museum when I was a kid. There is a Museum of Sports Art near downtown that I always want to check out, because the concept intrigues me, but I have never made it there yet.
Massachusetts Ave on the edge of downtown (near the Murat Theater) is sort of a mini-Short North with some shops and galleries. One of my favorite stores in the world, At Home in the City (cool, funky housewares and gifts and miscellaneous stuff), is located there. And they have some cool public art installations scattered about, so it's fun to just walk around.
If you are a dinosaur like me who still buys CDs, Luna Music is a pretty good indie store. They have a location on Mass Ave, but the one by Broad Ripple is bigger and better.