How long do you have?
May is the perfect time to just stoll through the downtown parks. End up at Forsyth Park and sit under the trees near the fountain with a picnic.
In to Civil War history? Fort Pulaski is on the way to Tybee Island. Take a tour there and then head out to Tybee for a walk on the beach and dinner at Social Club.
I usually prefer City Market over River Street so shops, bars and restaurants. Not as touristy.
Do you like craft beer and good food? I maintain a map of my favorite places. http://g.co/maps/85mws
Most folks lineup in the morning to get lunch at Paula Dean's. Skip that and do what we Savannahians do and get in line at Mrs Wilkes around 10:30 am and you should make the first seating. You won't go away hungry.
Railroad Museum. Revolutionary Park. Visitor's Center.
Our local Class A club, the Sand Gnats, will be playing at Grayson Stadium in May.
So much else. Let me know what interests you and I'll focus more on that.