I think that if you only go to one pizza place in Indy it should be Bazbeaux's (http://www.bazbeaux.com/
). Preferably in Broad Ripple where it originated. I've not been to the newer downtown or Carmel locations. They're probably nicer, but probably not as authentic. Bazbeaux originally brought the idea of upscale pizza to the area and clearly inspired other startups and are consistantly voted best in the area in many polls.
Also, although it's not a pizza place (as far as I know), you might also consider the Mug 'n Bun near 10th and Lyndhurst. I've not been there myself, but it's virtually a legend amongst Indy 500 fans (and many Indy 500 drivers). I'd classify it in the 'Diners & Dives' category. The thing to have there would be the pork tenderloin sandwich, along with the homemade root beer. The deep fried tenderloin thing is very Indiana.
If you'd like to combine those two Indiana experiences in another dive, the Pizza King at 1904 Broad Ripple Ave. This used to be the 'Tin Star'. A fried tenderloin joint with booths in the basement with barred doors - as though you're being incarcerated. Now the bulding as been bought by Pizza King, which is Indiana's local blue collar chain serving what I call Indiana Style pizza (I'm originally from Michigan, but this style has grown on me). Thin crust, tending in Pizza King's case towards the cracker style. And toppings to the edge in an attempt to eliminate the crust left on the plate. Apparently, there was enough of a backlash in the change in ownership that the Pizza King went back to the previous owners and got the Tin Star tenderloin recipe and are now advertising Tin Star Tendrloins on the Pizza King sign.