Conner’s Prairie would be way too far away without a car - it’s way up near Carmel.
St. Elmo’s has been mentioned, so I’d recommend hitting The Slippery Noodle, Indiana’s oldest bar. They usually have blues bands playing, and the atmosphere is great. Ask the waitress to show you the bullet holes from John Dillinger.
The Children’s Museum is very cool, even without kids with you. The Eiteljorg will be walking distance from nearly anywhere downtown (where I’m going to guess is where you’ll be staying). They used to have a very large Ansel Adams exhibit, which was very impressive.
The NCAA headquarters has a great exhibit as well, and well worth it.
If you’re a military buff, there’s plenty to see. The U.S.S. Indianapolis museum is downtown somewhere (infamous from the Jaws monologue of the ship that went down in the Pacific, and the crew was attacked by sharks). The War Memorial in Monument Circle is very impressive - there’s a free museum underneath it, and well worth an hour to walk through. Down by the canal is the Congressional Medal of Honor Memorial, which I was incredibly impressed with. I would definately venture down there to take a look.
If you eat at The Calladdagh, Zappo would probably recommend the fish and chips, but suggest that you get the 1/2 order. I concur, if not only for the chance to see the DopeFest-famous lemon condoms.
The Pacers won’t still be playing in July, but they might still be having practice. One of their practice courts has a public window, so passersby can have a look.
IUPUI (Indiana University Purdue University at Indianapolis) has the country’s fastest swimming pool - The Natatorium, where the U.S. Olympic team has their qualification meets. It’s walking distance from downtown as well.
The zoo is very good, and I’ve heard that the jellyfish exhibit (UFO’s - Unidentified Floating Objects) is pretty cool. I haven’t been, but I’ve only been here a little over a year.
If you’re still hurting for good restaurants, Indianapolis suprising has a surplus of excellent sushi restaurants. One of the new hotspots is Lotus, a sushi lounge/dance club/champaigne bar. They play Kung Fu movies 24/7 in the sushi lounge. I haven’t been, but everyone raves about it.