I love Indianapolis, and coming from Michigan, that takes a lot to say.
As to romantic things to do...not terribly sure on that, I go there for conventions. However there are some favorite restaraunts and such.
Bocca del Bippo- Italian restaraunt serving 'family' size portions...plates big enough for two (or more). Nice thing to share with a love interest. They also have an incredible chocolate cake/dessert thing with a chocolate liquer syrup.
The Ram- Great little bistro with its own mico-brewery.
There's a martini/cigar bar down there in the same general area (can't remember the name, it's something like a guy's name...help out anyone from Indy...like Martini Pete's or Cigar Joe's)
Edit: add:
The whole downtown area (by the football stadium) is a nice place to go for walks. Fountains, impressive buildings...there's a big mall/skyway thing that runs between stores and hotels that is VERY nice. And it's even safe at night.
AND there's carriage rides downtown.