K.M.
We moved to the Suwanee/Forsyth Co area (just over the Chattahoochee river from Sugar Hill) this past summer. What neighborhood are you moving into? Maybe there is a neighborhood mom's group? It has been a great way for me to make friends and for my kids to make friends that are close by. We LOVE our new neighbors and how active everyone is.
The libraries in both Gwinnett Co and Forsyth Co are great and have programs for the kids. There is an indoor public pool w/ slides and playground w/ squirt guns/water falls/tunnels to climb through not too far from sugar hill - Bogan Park. It may be in Buford but just 10 or 15 min from Sugar Hill. Swim Atlanta is a great place for off season swim lessons.
Monkey Joes and Catch Air are indoor playgyms with inflatable bouncers and slides etc. for rainy days.
Check out this webpage: http://www.suwanee.com/cityservices.recreationparks.php
There are several parks connected by a greenway along the Suwanee River and the play town suwanee is worth the drive. My 3 yr old calls it the "castle park" and requests it on a regular basis.
Hope that is helpful! Lake Lanier is in your backyard so there are lots of fun places associated with the lake as well.
Good luck with your move!