An eight mile canal runs through downtown Indy. At one end sits the Creation Cafe, and near the other end looms the Indiana State Museum. I went out on St. Patrick's Day to get a shot of the canal, because the water had been turned green. It must have worn off by the time I got there, though.
Several murals line the canal.
Some things to do: rent a paddle boat, walk, jog, ride a bike, visit museums, have lunch, or play near the fountains.
But if you're going to walk for long... bring some shoes meant for such an occasion.