I spent the first 3/4 my life in Indianapolis, and I can unequivocally say July would not be my choice month to visit. It's not Texas hot, but it's humid beyond measure. As far as natural beauty goes, well, Indiana does have trees which west Texas lacked. Indiana has worked hard to bill itself as more than a race track surrounded by a corn field, but hmmm I don't recall a whole lot else.
Definitely go to the Speedway museum, and take the tour bus around the track. It's a spectacle. Better on race day, of course.
Downtown Indy has a night life these days, if you're in to that. Broad Ripple isn't far from downtown for other entertainment options.
Check out the Major Taylor Velodrome schedule when it's closer, in case there's some kind of event. I've watched world class competition there, with a crowd of 75 people. I think these days it's more locals; they days of Nelson Vails, Mark Gorski, and Curt Harnett are long gone.
Downtown Indy has some decent bike trails now, if you want to do some urban exploring and don't mind being drenched with sweat after 2 minutes outside.
When I moved to Texas, I drove my Mazda Protege, two bikes on the back, music blasting, 4/80 A/C. In frickin' August. I took the Missouri/Oklahoma route that time which, I dunno, the Arkansas route isn't any more interesting. Which way are you going?