Definitely go downtown, see the "Circle" with the Soldiers and Sailors monument, take a walk on the canal (you can rent surrey bikes or paddle boats there), take in the zoo and gardens, go the Eiteljorg Museum or State Museum, there is an Imax theater in the state museum.....definitely eat dinner at St. Elmo's or Harry and Izzy's (they are owned by thte same people) both have their "famous" shrimp cocktail (it is SUPER hot!) and great steaks! They are pricey, but worth it!
If you have any questions please feel free to PM me, I am a Hoosier native, and worked for the hospitaility industry for 10 years in Indianapolis!