I'll put a plug in for you to visit the Sun King Brewery (which I am affiliated with) when you are downtown, and the Hampton is still sporting it's "new car smell" from being revamped as the Bud Light Hotel during the Super Bowl, so it is a good choice among many good choices. There are some really nice restaurants to try, but as ArmedRPh posted, if you have food preferences I know INGO will give you the skinny on some great choices. I really like to support local so my suggestions would skew that way, but you have all your usual national chain suspects available to you as well.
I will second rockhoppers recommendation about SunKing great beer! It can be found in seveeral places around Indy. Never been to the source, but I would love to.
If you are downtown and like a good Reuben sandwich, Shapiro's Deli can't be beat. Other great foods too.
Not sure if you are wanting good and expensive recommendations or more the cheaper but still good stuff. You want steak, seafood?
If a little from downtown is OK, Rock-cola Cafe is a great 50s diner.
La Margarita and/or Revolucion in Fountain Square (a few blocks away from downtown) would be my recommendations for Mexican. One of them is right across from the Fountain Square building (the anchor of the whole place), the other is just a few doors down. Much bigger menu at La Margarita, Revolucion is more small portion tacos and such, but both are very good. For Italian downtown, there are several ok choices, but nothing comes to mind that is really stellar - I'll let someone else suggest a favorite Italian place downtown. Not Mexican or Italian, but Ralph's Great Divide is a lesser known jewel, in my opinion.
I second the stay away from chains if you can. IMO, the best Italian in town is Mama Carolla's in SoBro (54th and the monon trail). Great new pizza joint opened downtown, Coal Fired Pizza. Super tasty. Only a block or so off the circle. I haven't tried any of the Mexican style places mentioned prior. Most of those that I have tried in Indy have been lacking. Great tamales available at the Tamale Place on west side. Have fun.
I know this isn't food or lodging, but if you have never been there, The War Memorial is an awesome place to spend a couple of hours. That Upper Room is phenomenal.
I agree with supporting local restaurants. I would suggest either Iaria's or Milano Inn for Italian, and El Sol De Tala for mexican. All are very convenient to downtown, but not necessarily by foot so you may avoid the large crowds going to eat there. Save the time on foot for going to the games and hitting the bars at night.
Tavern on South is good. on south street west of Lucas Gun Oil Stadium. Try the tenderloin sliders on pretzel buns (), and make sure you get some garlic aioli for your fries. Pretty sure they have Sun King on tap.
Hey, this is a pretty smart group (I agree on most counts):
Hotel - Hampton
Beer - Sun King, Wee Mac
Food:
Best Seafood in town at Ocean Air (but $$$)
Pregame either Scotty's (close to hotel and game) or Champs (in the mall)
It is not cheap by any means. But these people will throw some food on you like you cannot imagine. I layed in the fetal position for hours in a meat comma after eating at the Atlanta GA location.
I have also heard wonderful things about the restaurant next to the Conrad. I haven't tried it yet but plan to.
Weber Grill has some good stuff as well.
On a side note please do be careful walking downtown on the weekend nights. Not always a good place to be. It can get ugly quick.