Great news! Indianapolis has a great few courses, also PM me on here if you want any housing advice in the area.
In the Indianapolis metro area, there’s definitely more good private golf than there is public golf. With Pete Dye beginning his architecture career in Indianapolis it’s hard to find a course that hasn’t been influenced by him. If the broad ripple area is going to be home base, I’ll give you a few public tracks within the 10-15 mile range:
The Fort https://thefortgolfcourse.com/
This course was built on land that was once a military base. Pete Dye and one of his associates Tim Liddy built this course in the late 90’s. I wouldn’t consider it a walkable course, but it’s pretty fun. Unfortunately, the play can get pretty slow…
Brickyard Crossing https://www.brickyardcrossing.com/
This course is located right by Indianapolis Motor Speedway, holes 7-10 are actually in the infield of the racetrack. An LPGA tournament is played here every year. The greens fees reflect this fact. I’ve actually neglected to get out and play this course quite yet, so I feel a tad fraudulent recommending it but i’d be remiss not to include it.
The next two courses are a bit farther north than Indianapolis. There are a TON of courses in this area, but I’ll give a few recs of courses that I enjoyed playing recently
Plum Creek https://www.plumcreekgolfclub.com/
Another Dye design in Carmel. Not sure where you’re from, but if you imagine the most uppity suburb from your hometown, that’s what Carmel is. Tons of both public and private courses in the area, Crooked Stick is one of the most notable private courses in the area where John Daly won the 91 PGA as an unknown. However, Plum Creek is a public course with a pretty decent membership of folks that live around the course. I had a pretty good time playing out there the last time I was there. There are (2) other courses that but up against this property (Prairie View and River Glen), however this one is my favorite.
Golf Club of Indiana https://golfindiana.com/index.htm
Not much to say about this one. It’s a little north of Indy, but you feel like you’re on a farm. I had a pretty enjoyable walk the last time I played here.
If you’re looking for something more particular, or private course recs I can do my best. Like @ausmil said, Broadmoor is probably the only course I’d consider joining in the metro area. Donald Ross course and they’re doing everything they can do to get members. Waived initiation and low dues (at least the last time I inquired about membership).