If you like seafood, go to the Oceanaire Seafood Room, 30 S Meridian St, (317) 955-2277. They have some of the best oysters I've ever had.
If you like steaks, there's a Ruth's Cris steakhouse in walking distance from the Oceanaire.
During the weekend, the zoo is about 1 mile from downtown proper, as is an I-Max theater and the Indiana State Museum.
The Indianapolis Museum of Art has over 150 acres of gardens. It's about 2 miles from downtown.
For gaming fun downtown there's a place called Jillian's which is like a Dave and Busters. There are a bunch of bars and clubs in that area. There is also a great Irish type pub there call the Claddagh... If you go there, get the fish and chips!
If you like Basketball (I don't) the Pacers are playing the Nets on Friday at 7pm.
If you go out of the downtown area, up North a bit to the 50th st area, there's an area called Broadripple that has nice restaurants, nightclubs, etc. Usually a younger group there, but still fun.
It's a great little city, one I'm proud to call my home now.
2 M80s, 1 VP180, 4 QS8s, 2 EP800s, 4 Algonquins, 2 M3 VaSSallos