I grew up in the Raleigh area, though I moved away for 15 years and only just moved back a few months ago; I don't have an up-to-date complete knowledge of gyms in the area, but there are a few I can recommend and a few I can suggest steering clear of.
Triumph Gymnastics in Cary is the gym I'm currently at. I'll say up front: it is not a competitive powerhouse. If winning meets and aiming towards potential D1 college gymnastics are a priority, we're probably not the best choice presently.
HOWEVER, our head coach Kathryn is extremely dedicated to keeping the gym a positive and nurturing environment, prioritizing both physical and mental wellbeing of every athlete. I've only been at this gym 4 months, but I can honestly say it feels like one of the healthiest environments I've coached in. I also cannot say enough good things about the owners, who are absolutely delightful people.
(Note: if you do go to Triumph, be aware that Geoffrey isn't my real name, so if you ask for me by this handle people will have no idea who you're talking about)
About five minutes down the road from us (literally on the same road) is TAG. I don't know their current staff well at all, but their reputation is somewhat more competitively intense than Triumph. I haven't heard much negative about them, and the facilities are nice. Definitely worth checking out.
On the other side of town up in North Raleigh is Energy Kidz. Energy is a big successful multi-sport business, and has some of the best facilities in the state; however, they have a reputation as not being a great place to coach, and as a result they tend to have pretty high coach turnover. Worth checking out, but don't get too attached to any particular coach.
Raleigh School of Gymnastics is about a block away from Energy. I'm not sure what to make of them. Their facilities are unimpressive; despite this, they have a pretty good reputation as far as I've heard. They've produced excellent gymnasts, and have done so without developing a reputation (again, as far as I've heard) for being overly harsh in their training. Definitely worth a visit if you're on that side of town.
Don't know a whole lot about Oak City, but from my conversations with the owner she seems delightful, and I get the impression she has a very positive philosophical approach to the sport. Probably worth checking out if you're in that direction.
Sonshine Gymnastics has an impressive competitive record; however, they have a reputation for being very harsh in their training. In addition, they have an overtly-religious angle to their philosophy that never really sat right with me. Personally, I'd avoid them.
Peak Gymnastics in Apex.... I have direct experience with the owner and some of the core staff, and I strongly recommend that you avoid them like the plague.
.... and there's like twelve other gyms in the Raleigh area that I know nothing about.
Best of luck!