Indy has one of the worlds biggest Children's Museums (it's changed a lot since I last went, but i always had fun there as a kid). here's their website:
http://www.childrensmuseum.org/catalog/home.asp
they have several free days, including MLK day and Christmas Eve, and the first thrusday evening of every month. Admission has gone up considerably since I was there last, but it's worth the money if you stay the whole day.
The zoo and gardens are also really great. the web page is here: http://www.indyzoo.com/
they about the same cost as the museum in the summer, but cheaper in the fall and winter and still pretty fun. we went this january on a warmer day (oh, say 55ish) and had a blast.
there's also tons of art museums and an Imax and lots of live theatre and good restaurants too.
There's also a lot of dancing that's pretty reasonable priced. There's a place that holds regular swing dances, and a regular contra dance as well. I dont' know about the swing, but contra is really family friendly and super easy to learn. We've had everyone from 6yr olds to 86yr olds learn here in B-town.
some web pages for that suff:
http://www.indycontra.org/Home.php
http://www.indyswing.org/html/schedule.html
hope that helps and welcome to the hoosier state!