I would second or third or fourth the recommendation of the Triad. Iām from WS and have lived in Raleigh, Durham, Chapel Hill, Burlington and now Iām in Greensboro.
For me, the traffic and lack of public golf knocks Raleigh off the list. Durham is funky, which I kinda like, but public golf outside of a few meh courses is lacking. The Neech notwithstanding @Double_Bogey_Dave of course.
Chapel Hill stinks, unless youāre a tarheel.
Drive as fast as you can past Burlington. I would shout out Alamance CC though, donāt turn down an invitation to play that Donald Ross design.
Greensboro has about a dozen public golf courses, only a few of which are subpar. Bryan Park, Forest Oaks, Jamestown muni, Oak Hollow, and Holly Ridge are my personal favorites, all within 20 minutes of downtown. I would include Grandover but @Fluff might self-destruct. I do practice over there quite a bit and find the pro shop folks to be incredibly hospitable and welcoming.
Of the private options in Greensboro, Sedgefield is my favorite course in the area, but the membership numbers are making tee times on the weekend pretty tough to come by (I live nearby and have a bunch of buddies who gripe about it). The McConnell connection is very tough to ignore though. There are a couple of members on the refuge, DM me if you want their info. Starmount is more affordable and has a younger vibe. Decent golf course with a few quirks. Greensboro CC can be a bit stuffy and could be hard to break in for a new, out of town gui. Fun but tighter course close to downtown, lined with large trees and older stately homes.
Winston-Salem has a handful of public courses (Tanglewood, Reynolds Park, Oak Valley, Meadowlands), but the affordable clubs are what drives the golf around there IMO. Maple Chase and Bermuda Run are two legit clubs with very affordable packages. On the pricier side, Forsyth CC is similar to my Greensboro CC description above. Old Town Club is probably the best golf course in the state outside the Sandhills, but you better come with deep pockets.
Iām sure I left some places out, but these are my takes early on a beautiful Saturday morning.
Good luck with the pending move!