Welcome to St. Louis - Now rent a car!
I've lived here since I was six months old, so I know the area as well as anyone. The first piece of advice I'd give anyone visiting St. Louis is that you will need to rent a car for convenience and in order to see most of the sights in & around our city.
We do have the Metrolink rail system, but it is fairly limited currently, and any expansion plans are pretty far off in the future. Plus, many of the best things to do and places to visit in St. Louis are located in the county and farther away from Metrolink. The Metro bus system does cover the St. Louis City and St. Louis County, but it's not the best bet for tourists.