Civil War museum in the basement of the big monument in the circle. I was down there today, and saw an authentic brass receiver, Henry .44 lever gun.
There's a Dr. Who museum, with limited hrs of operation. IM me, and I'll try to get details if you want them.
Cool Cafe Racer type bike shop, in Broadripple, called CityMoto.
If somebody tells you about gunshops, laugh in their face if they suggest Don's or Pop's. (unless it's somebody whose ass you should be kissing, in which case, come up with a good excuse to avoid those places) Everyplace else, have the batteries in your BS meter charged up.
cool canal downtown ETA: that's where you'll also find the aforementioned USS Indianapolis memorial
monument in the place Bobby Kennedy was speaking, on the day MLK was assassinated
if you dig public art & architecture, you can hit Columbus, IN, about 1 1/2hr south, on I-65. the whole town is cool to see.
Slippery Noodle, a blues bar downtown, with live entertainment 7 nights a week. IIRC, John Dillinger is rumored to have taken target practice out back. Dillinger is buried in Crown Hill cemetery.
Don Shula (Miami Dolphins) owns a fancy restaurant downtown.
Something like 70% of all insulin is made here
ETA II: Hinkle Fieldhouse, on the campus of Butler University, if you're a fan of basketball, and its early-mid 20th century history
Mann's Grill, if you like greasy spoon joints, phenomenal portions, and chunky waitresses wearing fuzzy slippers.
that's it off the top of my head. have any particular interests?