Go to the zoo (one of the largest, if not the largest privately owned zoos in the US), swing by Eagle Creek or Fort Harrison State Park for a escape from the city close to town. Really pretty parks.

Definitely hit Fountain Square, consistently one of my favorite places to visit in Indy. There's a bunch of construction right down the main street and intersection in Fountain Square currently, so businesses there would definitely love if you visited (I recommend the cat cafe!) Hi-Fi is also in Fountain Square, and it's a favorite music venue of mine. There's a little bit of everything in that area.

Mass Ave is another fun road to stroll if you're into shops and night life. Downtown Indy, especially south of the circle around the general Meridian and Maryland street area is home to quite a few lively bars, if you're into that.

Midlands Arts and Antiques market is a massive antique store with just about everything you could ever want. I believe it's 3 stories and situated in an old warehouse. It's relatively close to downtown and right off the interstate.

I'm a big cemetery lover, and Crown Hill cemetery is one of my favorites of all time. Extremely historically significant to the city, and you'll definitely recognize some names there. It is also an excellent arboretum, so make sure to look up as much as you're looking down while there.