I'm from Savannah, but I don't live there anymore. I do come back and visit a fair amount (I think the next time I'll be back for a while is in October). I get a lot of good riding in when I go back home! Have you checked out the group ride at the Gallery Espresso on Saturday mornings? It's at 8am, and they have split the group into 4 pace/distance options. It is also a no-drop ride (though I would do the "B" ride, because I wouldn't want to piss off the A riders hanging on for dear life--I do that on the Sunday hammerfest ride instead).
You can also find some good casual, long rides in the area through the Coastal Bicycle Touring Club (they have a website).
As for riding around town, HAAF is a good, safe place to go during the week or at higher traffic times by yourself. There are plenty of roads out of town through Garden City out around the airport, Pooler, etc. to ride on from downtown that are fairly quiet. Many of them are numbered Georgia Bike Routes. The people who do the Saturday rides are a really good source of information about where to go and when.
What kind of training do you want to do? Distance? Pace?
When I go home, I tend to do a lot of group rides and suffer, because all the veterans and superfast racers really challenge me, but they are supportive and do help me improve a lot. However, I can see how that level of intensity isn't really for everyone, but maybe I can help you get to know the right people for your goals...