There are different places within 10-40 minutes depending on how far you want to drive.
Eagle Creek is the go to on the West Side.
Fort Harrison on the East Side has some good running trails.
Washington Township Park in Avon is decent for trails.
The towpath and canal is solid.
The B&O is good but it's just your typical concrete path that used to be a railroad.
The Monon is like the B&O but closer to Broad Ripple and downtown.
DePauw Nature Park Trail in Greencastle is a nice place as well. About 45 minutes from the West Side of Indy.
And then Brown County has lots of hills and trails as well.
There's also a few nice cross country courses at the high school level that people train on as well.
I don't specifically know of any running groups because I've never seeked them out. I'm sure there are also plenty of places to run that I forgot off the top of my head.