Savannah is perfect in winter - soft breezes, humidity from the ocean, warm weather. Spring and fall are nice too, but skip summer if you want to gt your long runs in without melting.
Skidaway Island State Park is only 15 miles southeast of the city, and it's a really nice getaway for camping, hiking or mountain biking.
There are no real hills or mountains anywhere near Savannah - just low marshes, woods, and grasslands. You'd have to drive about 5 hours to the Appalachians to do any real mountain hikes, though it certainly does get hillier further inland in Georgia
Running through Forsyth Park looks like it might be fun, but it's not that big - do it once or twice, or maybe make it part of a run through the whole historic district.