Auburn is a big rival but probably not the biggest, though that depends on the age of the fan. UGA and Auburn have had a number of epic games through the years, and each has knocked the other from the ranks of the unbeaten numerous times. Then there is the Vince Dooley and Pat Dye cross over. Supposedly after the 1980 season, Auburn offered Vince a million dollars (probably over X years) to coach and his kids were dead set against it, which gave the bumper sticker "Some people wouldn't go to Auburn for a million dollars!".
Obviously, Florida is a long term rival and Georgia Tech has been as well. The latter is not so much for younger Dawg fans. UGA and Tech have also had some epic games but Tech obviously has not been as consistently good for a while.
UGA and Auburn play every year, while Alabama plays only on the rotation, which means once every 6 years on average. Yuck.
While Georgia is known as a state with fertile recruiting, you also see a lot of battles for recruits with Auburn and Clemson and FSU. I looked up the last four years 247 recruiting rankings and UGA holds an edge over OU, though I think that matters little considering Mayfield was a transfer walk on. UGA has usually recruited at a pretty high level.
I'll be a bit sad when Michel and Chubb graduate, but obviously wish them great success at the next level. They have some great life stories even aside from football. Chubb I think is better than many realize, especially if they looked at him before he was injured and then this year again. When Chubb stepped in for Todd Gurley, I thought Chubb might be better, as a freshman. Michel is almost the ideal for the NFL in terms of size and assets.
Next should be to get a home and home series with OU. I know OU fans are very hospitable in reputation.