Acapulco Joe's in the near downtown area is a great place to eat
I live here in Indianapolis - have all my life really....I second this little place.
Unfound - Its hard to say without really knowing what you like to do and when you will be coming to town.
If you are a history buff - then Connor Prairie would be a neat place for you. www.connerprairie.org
Into art? The Indianapolis Museum of Art would be great. They have in door and out door exibits (100 Acres is their out door exibit) www.imamuseum.org. As someone said before, the Eiteljorg Museum is a wonderful place. Right now they have Ansel Adams showing there as well. www.eiteljorg.org
If you like military things? We have the Indiana War Memorial. www.in.gov/iwm. The circle is actually the Civil War museum inside The soliders and sailors monument.
Interested in a show? We have comedy clubs, symphonies, and theaters. Need something bigger, we have 3 Imax theaters.
Need something to quench your thirst? There are wineries and microbrew pubs all over the place.
If you enjoy beer, different and/or international beers. I would strongly suggest that you take a cab (i have a cab guy you can call) and head down to Greenwood and go to Shallo's Antique Resturant and Brewhaus. They have around 500 different kinds of beer (between tap and bottles) and its amazing. It will be crowded. It is a very hole in the wall place.
Shoppting you say?? Well, the Keystone Fashion mall will quench your high end tastes with places like Coach. There are also things like Banana Republic, Gap, and if you so need, and Apple Store. (The Fox and Hound Bar and Grill is up there and I really enjoy that place).
We have food from every culture somewhere in town. If you let me know what you are interested in, I can do better justice to it.