Although St. Louis is a wonderful place to visit, you will want to do some logistical planning before you leave - tourist sites are spread around town, there is no quick and easily solution for vising the sites without a car and there are some neighborhoods that are, let's just say not so wonderful.  For planning, use Trip Advisor forum for advice on getting to specific attractions.  

St. Louis has a light rail system called MetroLink. It runs from Lambert Airport to Downtown St. Louis and then on to Scott AFB / Mid-America Airport in Illinois. A new line also recently opened, providing service into Clayton and beyond.  It is safe and cheap; free around downtown. Metro also operates several shuttles from the MetroLink stops to various attactions.  The bus system is also good and taxi's are readily available. If you plan to explore St. Louis neighborhoods you should have a car, traffic is not bad even during rush hour. Parking is usually plentiful and inexpensive, but at night, especially in the city, try to stick with well-traveled areas.