Well, others have already mentioned many of my own favorite Indy/Bloomington places, so I'll just try filling in a few gaps.
If you like to play Irish trad music, the weekly Tuesday night session at the Golden Ace Inn (2533 E. Washington St) is a place you should go. Bloomington also has a nice Irish scene, which I'll describe when we come to it.
The Slippery Noodle, as already mentioned, is the place for blues. The Jazz Kitchen, on College Ave. at about 52nd, is another favorite spot for good live music.
The arts districts in Indy, really, are Massachusetts Avenue downtown and Broad Ripple (around 62nd and College). Many galleries, funky shops, tiny playhouses, small excellent restaurants, and nightclubs in both locations.
My favorite downtown coffehouse (The Abbey) and pizzaria (Bazbeaux's) are on Massachusetts, which is only about a mile long. If you happen to be of that persuasion, Indy's biggest gay bar (The Metro) is about one block south of the Abbey. ;>
If you're up in BR, the Broad Ripple Brewpub has very nice beers, and is actually a local microbrew, as opposed to some of the national chain outlets (Rock Bottom, Alcatraz) that are found downtown. Bazbeaux's pizzaria also has a location in Broad Ripple. The big Indy nightclub (The Vogue) is there, too, on College just below 62nd.
The Herron School of Art's gallery is at the main campus, 16th and Pennsylvania. The Indianapolis Museum of Art, which gets many fine traveling exhibits, is at 38th and Martin Luther King (its name below 38th; above, it's called Michigan Road).
Tons of arty stuff in Bloomington. I'm mostly down there for the Irish music, which includes a weekly Sunday night session at the aforementioned Irish Lion (5th St. just west of College) and a first-Friday-of-the-month session at the College Mall Border's hosted by Grey Larsen, an excellent and well-known flute player.
Old-time music buffs have a weekly session in the basement of the Cellar Lounge, one block south of the courthouse square on Walnut. The Cellar features lots of other types of music as well.
There are so many places to eat a good meal in B'ton, it's hardly worth mentioning any. However, if it's a really good burger you're after, it's Opie Taylor's you want (just north of the courthouse square on Walnut).
If nothing else brings you to B'ton, go down for the annual Lotus folk music festival (somewhere towards the late summer/fall, I think). Always a wide range of acts worth seeing, at prices ordinary humans can afford.