Scooters are definitely awesome - easy to park, etc. If you don’t want to take them on the open highway, you can easily arrange a taxi ride to Tybee Island for a taste of the beach, pier, and lighthouse (which I do recommend: It’s a fun and funky little island. Try the North Beach Grill while you’re there.) And there are also free shuttle buses around the downtown area, and a free water taxi across the river between River Street and Hutchison Island. (Even though there’s nothing much on the island, it offers a nice nighttime view of downtown, and the water taxi ride is fun, quick, and free.)
Downtown, I also agree with the restaurants recommendations so far. Also, Moon River Brewing Company on Bay Street has decent food and some pretty nice brews. I like Vinnie Van Go Go’s in City Market for a slice to be eaten outside on the plaza - you should have near-perfect weather for your visit, barring rain. Jazz’d Tapas is fun for dinner and music. Also, check out the rooftop bar at the Bohemian Hotel, overlooking the river.
Other than Bonaventure Cemetery, Tybee, and the squares, I’d recommend the following (in no particular order): Telfair and Jepson museums, Railroad Roundhouse museum, many of the house museums (Greene-Meldrim house, Juliet Gordon Low and Andrew Low houses, Owens-Thomas, etc.) and the museum at the Savannah Visitor’s Center.
Actually, I often recommend that first-time visitors take one of the on-off trolley tours of the downtown district. That way, you get a good overview of the area, hop off at spots you’d like to look at more closely, and spot places you’d like to explore further later.
The Starland district (just north of downtown proper) has some funky shopping, and there are lots of art galleries and such throughout downtown.
And Balthisar, have you considered staying in a hotel a bit further afield? I can recommend some pet-friendly hotels 5-10 miles away from downtown. In downtown itself, though, I know that the East Bay Inn is well-located and dog-friendly, along with the Thunderbird (retro-funky restored fifties motor court,) the Holiday Inn/Mulberry on Bay Street, and the Comfort Suites Historic District (which is a little west of downtown, actually, in a neighborhood that’s slightly sketchy after dark.)