Savannah is a beautiful city with a rich history. I live an hour from downtown and visit often. The carriage tours are pricey and each tour guide tells a brief history of the sites along the way. They are great if you can afford them if not there are walking tours. I have bought several tour books and walked routes myself and that is fun as well.
There is way too much to do in one day but I recommend a visit to Forsyth Park, Colonial Cemetary and any of the Squares, they are all beautiful and full of monuments and history. River street is really nice, a little touristy but in the evenings, especially weekends there are street musicians and sometimes dancers and artists. The Pirates House has good food, a bit pricey and has a mini tour inside, great if you like the ghost stories. The Ghostly side of Savannah is fun and the hearst rides are a riot, the best tour we took!
There are museums and historic sites all over and people are always willing to share a tale or two so just ask.
When on a budget there are hotels mid town if you have a ride to get to the downtown area or try priceline for a good deal. There are often conventions and times of the year that things do get booked so be aware of when you want to go and when things start picking up for tourists. The savannah tourism website often has specials and deals so look there too.
When on a budget pack a lunch and go to the park and visit one of the many coffee shops for a decadent dessert. Some of the souvineer shops have nice things that are not very expensive. Also there are places to park that are free, really. Ask a local for tips. People are generally pretty friendly. As long as you don't mind walking, and that is the best way to see Savannah is walking, you can park for free and see the city too.
I used to live in Charleston SC and it too is a gorgeous city. It is more expensive though and parking is even more difficult but if you can do both it is well worth it. My teens actually let me tell them about our country's history and the role the city played in it and that itself was worth the trip. I guess sometimes seeing and reading about it is two different things.