I mean, I will be really honest. St Louis is getting better as far as public transit goes... but we are not great. Our metro link is extremely isolated. Locals don't really use it unless they work downtown or live in the city and work at UMSL or the airport. Our bus line is somewhat inefficient unless you really know what you are doing. The metro link runs further north to east. It is not city/county wide like cities like Chicago. You cannot get around a town like St Louis without a car, unfortunately. Webster Groves is WAAAY south of the Metro.
Downtown to Webster: only real way is taxi. The metro link does not go that far south. Downtown to Webster... probably talking $25-30 by cab. Just make sure the cabbie takes highway 44, not 40. More direct to Webster (cheaper)
You could probably swing something else... but trust me... it would not be worth the headache. Do yourself a favor and cab. Forest Park metro station is only like 5-10 minutes from downtown. By the time you get off of the highway and navigate to the station... you could have been at your hotel.
Sorry, we are getting better��� just not there yet. At $2.20/gallon��� wish that we were. =)