Savannah's Troup Square is arguably the city's most popular see-and-be-seen park for people and pups. This leafy-green square is bordered by beautiful brownstones and row houses, and has plenty of benches for breaks. Leashed dogs are quite welcome here and at outside tables at several nearby restaurants (try brunch at Firefly Café, which borders this park). The best feature would definitely be the cast-iron water fountain just for dogs with two low-slung bowls for lapping, plus a special refilling mechanism.
Also noteworthy in town is Paws on the Island. Bring your pooch for their state-of-the-art hydrosurge spa baths to minimize stress and maximize cleanliness or to their soundproof "quiet rooms" for an aromatherapy treatment, and pick up some cute pet gear from their retail shop as well as a special treat to end the day!