First off, happy anniversary.
As far as food, a few options depending on budget.
If you've got the time & money, St. Elmo's is the iconic local steakhouse. It's downtown (also has a couple of locations known as Harry & Izzy's - same restaurant, same menu, but one is downtown and one is in the north suburb of Carmel).
I'm a big fan of Italian food. Iaria's - located just east of downtown - is the best Italian place in the city.
Charlie Brown's in Speedway is a treasured local place. Also around the corner from the Indianapolis Motor Speedway is the Mug 'n Bun. They're known for double-decker cheeseburgers and root beer, but it's a fabulous little drive-in with carhop service.
For breakfast, you can't go wrong with Cafe Patachou. There's a location downtown, one on the north side in Broad Ripple and another in the north burbs in Carmel.
(Also for breakfast: Long's Donuts. They're legendary here locally).
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Things to do:
I highly recommend the Indianapolis Motor Speedway Museum to any visitors. Even if you're not into racing, you'll appreciate the large collection of classic cars and racecars and the history of the track. Make sure to take the track tour - it's one lap around the IMS oval on a bus. You'll gain a real appreciation for the history and sheer size of the track. Price is very reasonable, too.
If you're going to the zoo (which is a favorite - it's fairly compact and right downtown), both the Indiana State Museum (largely a state history museum) and the Eiteljorg Museum (which is a Native American art museum) are fairly close. The NCAA headquarters and its Hall of Champions museum (which is OK. It didn't really overwhelm me the first time I visited) are also fairly close. A stroll along the adjacent canal or a paddleboat ride is also a favorite activity for me & the wife (and our kids).
The ECHL's Indy Fuel also have a pair of home games next week - one on Memorial Day and one June 2 (the regular season wraps up next week), although I think the Bruins will be playing both nights. I'm the play-by-play announcer, so if you come, feel free to stop by the booth and say hi before warmups or after our postgame show is done. The Triple-A baseball team (Pirates affiliate) has a home series next week. Their ballpark has a great view of the downtown skyline and has consistently been rated one of the top minor league parks in America since it opened 25 years ago. I believe both are playing to 50% capacity right now.