Fountain Square has arguably the best dining experiences in Indy.
Milktooth is hand's down the best restaurant in Indianapolis, and is consistently awarded top restaurant status world-wide. They are a not-so-average breakfast joint that is worth the time invested. Enjoy the coffee bar while you wait -- I believe they strictly brew Intelligentsia coffees. $20-30/person.
My personal favorite restaurant in Indy is right across the street: Bluebeard. They are a lovely Kurt Vonnegut-theme restaurant and bar with farm-to-table dishes and a great house-brewed ginger beer. Sunday afternoons, they offer $1 oysters that move quickly. $20-70/person.
Pioneer is on Fountain Square proper, and is very similar to Bluebeard. I believe the head chef and/or owner use to work at Bluebeard, in fact. They have great plates for sharing and a good enough beer/wine selection. The beef tartare is simply amazing. $30-70/person.
Thunderbird is a good option for a late night meal. They have a decent beer menu and good enough cocktails to pair with chicken & waffles or poutine. $20-40/person.
Kumas is one of the better burgers you'll find in Indy. After opening shop in town, they later closed one of the two in Chicago. Friends from Chicago say that the Indianapolis shop is better than the original. Great beer selection, even better burgers. $20-30/person.