Where you should live depends largely on where you work, your desired lifestyle, your interests, and how much you're willing to pay in rent.

The most popular young, trendy areas of the city include Downtown, Broad Ripple, Fountain Square, Near Northside, Castleton. Downtown is obviously the most urban but it's also the most expensive, especially with a car. Broad Ripple is semi-urban village with a mix of college students and yuppies -- it's also expensive. Fountain Square is pretty urban and a mile from downtown -- it's also the most diverse and has a bit of an artistic vibe. Near-Northside is a dense historic district that's rapidly gentrifying with tons of development between College Ave and the Monon; it has a few bars/restaurants of its own and it's not too far from downtown. Castleton is much more suburban but with tons of reasonably priced apartments; it has malls and a state park nearby and great highway access for anything else the city offers.

Unfortunately they're some of the most expensive areas in Indy but a few can be reasonable, especially with a roommate or two (particularly Near-north or Castleton). There's also many more options if you don't need to be in a young trendy area.

u/ExIsFu also has some great information.