Skidaway is possibly our favorite state park campground. Sites are very large, and the Live Oaks draped with Spanish Moss make it very attractive. As for things to do in the immediate area, there isn't a lot. As JayGee mentioned, there is a good grocery within a mile or less, turning left off of the park road. We didn't find this any sort of distraction though, as the drive into Savannah and/or Tybee Island is not too long and the quality of the camping more than made up for it.
As for things to do in the area, Savannah is a great city to visit with all kinds of things to see and do. In this case, Trip Advisor is your best friend. If you haven't downloaded the app onto your Apple or Android device, get it. It's free and chock full of good information. Having said this, a few of the things we enjoyed were...
1 - Shopping on River Street. Fun places to browse, really good restaurants, and you get to see close-up the huge container ships coming down the Savannah river towards the port.
2 - Walking through the many squares spaced throughout the old city of Savannah.
3 - Consider taking a food tour. These are guide-led 2 or 3 hour walking tours between 5 or so of the best local restaurants in town. You'll get an appetizer-sized portion of food at each establishment, along with a bit of the history of the town during your walks. When you're done with the tour, you will NOT be hungry, and you'll likely find a restaurant or two that you'll want to revisit on your own. Cost is in the area of $45 to $50 per person.
4 - Don't know what the weather is like on Tybee Island in December, but if it's warm and you head in that direction, but sure to stop at the Crab Shack for lunch or dinner. Seating is outdoors in a really kitschy setting, so do this only if it's reasonably warm. There may be some indoor seating as well, but I'm not sure of that. The food is really good.
If/when you drive into town, be aware that parking can be a challenge. We ended up parking in one of the downtown parking garages which can fill up very quickly during the weekdays. Cost is around $15.00/day.
Hope this helps. Savannah is one of our favorite places.
Enjoy... Bill
2008 Jayco Designer 35RLSA pulled by a 2007 Chevy 3500 D/A SRW