Once my boys are a little older and potty trained, we will make trips to Athens a regularity.
I gotta tell you the same thing I tell everyone I know that's an alumnus with kids: come to Athens during fall football season. The tailgating environment is, in my opinion, great for kids. My buddy brought his 3 and 5 year olds for Homecoming - the whole deal - parade, tailgating with hot dogs, game and halftime band show, and they loved it. Now, they had to leave early 3Q because the kids were done, but they obviously enjoyed the day.
I've also brought my buddy and his kids for basketball - and they're not even Bobcats - and his kids asked when they could come next. Athens makes an impression on kids, and despite the memories we have as students, going to bars and parties, etc., Athens is a great place to bring the kids for the day. And other than money spent on fancy Jackie O's beers, Athens is an inexpensive place to spend the day.
(you probably already know all this, but it never hurts to remind, or put it out there for others)