My golfing time in Arkansas was mostly limited to ages 13-15 (I grew a foot in one summer and the wheels came off) and didn’t pick the game up again until I landed in Chicago five years ago, fwiw.
Public: If you want some truly local flavor, play Brush Creek Golf Course. It’s a little country 9 holer where I learned to play. The course is set in a cow pasture with a prominent ridge and creek. The guy who takes your money at the register is the same guy who built and designed the course. Jeans and t-shirts. BYOB. $10 for 9 holes. It’s pretty special, if only to me. I played it over the holidays this year and had a blast.
Private: Pinnacle CC, in Rogers, holds an annual LPGA event; Fayetteville Country Club is listed as a Perry Maxwell design; Blessings Golf Club is an RTJII design that’s supposed to be brutal. I haven’t played any of the private courses but they are generally accepted as the “best” in the area. Although, I did have a fine meal in the Blessings Club House (designed by one of my former professors).