I live outside of Atlanta, and I can tell you first hand, the Georgia Dome is not fan friendly for tailgating....AT ALL!!!! I'd go so far as to say don't even consider it!
Official Dome parking is very limited, and most is usually reserved for VIP's, officials, sponsors etc. The rest of the parking is 'gypsy' lots...ratty old lots, people's front yards and the like.....and VERY expensive!
The only option, really, is an area the locals refer to as "The Gultch". It's hard to describe, and even harder to give directions to! But, basically, it's a railroad gultch area located off Marietta street. One lot is under the street/CNN-Philipps Arena, with an open air lot that joins up to it. As game time gets closer, and your serious about tailgating near the Dome, I'd look into it. It's your only real option.
And as stated before, when you go to the game, don't leave anything out you don't want gone when you get back. Homeless people in that area are thick, and they'll steal the paint off your vehicle if you arn't careful. I don't mean that in a politacally incorrect way, but TRUST ME!!!
IMO, you'd be much better off from a money/security/convience standpoint to take MARTA into town. It's not perfect by any means, but it is a good alternative. Getting to the game is easy enough, leaving the game can be VERY tough, with such a large influx of people hitting the system all at once.
The good news is, there's PLENTY to do down in the Dome area. CNN is near by (which has a food court), a NEW Taco Mac is by Philipps Arena (opened just this week), the Olympic Park (a GREAT place to just hang out and chill), the Aquarium, the new World of Coca-Cola just opened (located next door to the Aquarium), and there's a sports bar called 'Stats' near the Dome as well (but it's a bit on the small side).
Sorry this is so long! I'm a STH for the Thrashers, so we spend a good amount of time down in that area. As the game gets closer, feel free to PM me. I don't 'know it all' about the area, but I'll help out if I can!