Apex, Jones Park, Wendell(last 2 years) opened recently. Apex has had some work done to it this year so I was told last night. Wendell still needs some work. Hole one from the pro tees is a luck shot. Jones Park is shaping up nicely near Middlecreek in Holley Springs. Buck Horn has been around for years, nice course south of Raleigh in New Hill. UNC is another nice course in Chapel Hill. Valley Springs, in Durham. Zebulon has a nice course. Pittsboro has a nice course. There are a number other that I did not list.
At last count there was around 25 courses when you include public, corporate and private courses within 25 miles of the city limits. Some privates and corporates don't show up on DGR at this point.
Three clubs to look up are TCP Disc Golf Club in Raleigh, Chapel Hill Disc golf club and the Durham club can help you find club events. There may be others that I am not aware of at this time.
TCP has glow running through Feb, Durham meets either Saturday or Sunday morning for doubles year round I do believe. can't tell you about Chapel Hill during the winter months but they run a nice summer program.
Welcome to the area.