If you are interested in cemeteries with old tress check out Bonaventure Cemetery
https://www.bonaventurehistorical.org/ Likewise there are many beautiful old buildings in downtown Savannah as it was not burned during the civil war. Check out the area around River Street and Factors Walk. There are really interesting stone ramps and alleys that are really cool at night. The area around Jones Street have really great street views with the old trees hanging over them. Spend a lot of time just walking around downtown.
Eat at Mrs. Wilkes
http://mrswilkes.com/ for lunch and Olympia Cafe or the Old Pink House for Dinner. I hope you are there for more than one night as there are a lot of really great restaurants in Savannah. If you enjoy a good cocktail or two Savannah does have open container laws, meaning (North of Jones Street) you can walk around with a drink as long as it is a plastic cup. Given the number of bars and restaurants in Savannah a great way to explore is get a drink "to go" and walk around, when it's empty find another bar or restaurant and get another drink to go.
-Joshua
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