Jerry,
Welcome to the area! I've lived in Chapel Hill for eleven years and am a member at Governors Club, a pretty decent Nicklaus course with 27 holes. We're about 30 minutes away, and currently have a pretty incredible membership drive under way, but the junior program is a best limited with not much participation, although they've tried. We do have a First Tee program here, so hearts are in the right place on this one. This wouldn't be my first choice living in Cary, but the place is beautiful and might be worth your seeing.
The best junior program anywhere in the area is the Carolina Country Club in Raleigh. Ted Kiegiel is the Director of Golf, and he's built an incredible program. Webb Simpson grew up there, and a number of seriously fine young players are there now. But it's not a mill for generating just top players. Go out there when there's no school and you're likely to see anywhere from 10 - 30+ kids of all ages on the range, putting green, and course, and everyone really digs them being out there. It's a great club in other respects as well. The big downside is that there's a waiting list, but that tells you something about the club in this type of economic environment.
Raleigh CC is worth checking out. A solid Donald Ross lay-out, and a very family oriented club that was the first club purchased by John McConnell. If I lived in Cary or Raleigh that's probably where I'd join.
Old Chatham might be worth investigating. It's purely a golf club, and they are developing a wonderful golf culture there. The recent past president of the USGA is a member there. I'm not so sure about junior element, but it might be something they are trying to build. Certainly worth asking.
Check out Prestonwood. It's in Cary, and it has three courses. It's a massive club, developed primarily for SAS employees but open for membership to non-employees. Given the nature of its membership, I'm sure there are kids all over the place there. They hold a pretty popular Senior Tour event there every fall.
There are some other clubs in the area, but this should get you started. Hope this helps.