I just moved to Savannah in July and I love it! There is a ton to do and even just walking around the city, doing nothing is a great way to spend an afternoon.
I second Olde Pink House for a nice dinner out. Vic’s is nice too but there are also some nice lesser known places such as Sapphire Grill, Alligator Soul, Cha Bella, 17Hundred90, or B. Matthews which are just as good.
As for lunch, if you are going to do one Southern lunch I would do for Mrs. Wilkes. It is better than Lady & Sons by far and a really cool atmosphere. The lines can be extremely long and exhausting at both, especially during August, so I wouldn’t bother trying to do both. There are some other fabulous lunch spots in Savannah that are much more worth it includnig Soho South Cafe on Liberty Street (it’s an art gallery and lunch place, really neat), Zunzi’s (for a quick lunch, no inside seating, but delicious), and B. Matthews on Bay Street (great anytime of the day).
If you’re looking for breakfast or brunch, J. Christopher’s on Liberty Street is good, B. Matthews also has a good breakfast, and Clary’s is a greasy spoon but delicious (it was made famous by the movie, MIdnight in the Garden of Good & Evil).
I don’t know if you are looking to have a more relaxed dinner with pub/ bar food but The Distillery, Crystal Beer Parlor and Churchills are all bars that have pretty good food too.
As for places to go, either a trolley or carriage tour is a great way to get oriented with the city. Tybee is nice if you want to go to the beach and Fort Pulaski, on the way to Tybee is a good stop. City Market is a fun place to hang out, if you go on a weekend there is always live music. I second what someone else said about walking down to River Street but don’t eat there. It is very much a tourist trap with little good shopping other than trashy junk stores and nothing but fried seafood places.
If you’re into art, the Telfair Museum or the new SCAD Museum of Art are both great. Honestly, I did the Juliette Gordon Low house and it wasn’t really worth it but we may have just had a bad tour guide (I was a Girl Scout all my life too).
If you have a chance to make it down to Forsyth Park, it’s really beautiful and has a gorgeous fountain with moss covered oaks leading to it… exactly like you picture in the movies.
If you have other questions, feel free to message me. I haven’t been in Savannah long but I’ve had tons of people visit so we’ve done a lot!