St. Louis is not a friendly town to those without cars.
The metrolink will get you from the airport to downtown and hit a few major points in between (the stadiums etc.) but should not be relied upon by tourists unless said tourists are prepared for a lot of walking (and with St. Louis humidity that walking can be quite unpleasant). Be prepared to limit what you can see unless you are willing to pay for cabs.
Back to the city... Go to a Cards game (it is amazing to see how much of the city's psyche is invested in that team), go to the brewery, get some pretzel's at Gus's next to the brewery. If you enjoy walking hit forest park and the free zoo there. Check out some of the cool old neighborhoods like the Soulard and the Central West End. If you are in to nightlife, the Washington Ave. area is nice and close to downtown. Tourists do nightlife down at the Landing (on the river); locals do nightlife in the various neighborhoods (Soulard, CWE, the loop, Clayton, etc.) locals can tell you what vibe predominates in each area.