I'm an undergrad at slu right now, so if anyone has any specific questions, just let me know.
But I agree with the others--stay with a host. There is nothing really to do right around campus, but its not too far to the things others were talking about. There's lots of free stuff (zoo, art museum, history museum) in Forest Park.
Metrolink ($3 single ride from the airport, $1.25 single ride elsewhere, or $4 one day pass) runs from the airport right to SLU, so you don't need a cab as long as you don't have a ton of luggage or anything. The SLU metrolink point is in the middle of a bridge, and there's shady-looking people there most of the time, but I've still walked there and felt plenty safe. You can just push a call button, and someone from SLU department of public safety will pick you up and at least take you the couple of blocks to the med school. From there, you're probably on your own, but I bet most students have a car, unless they live right by the school. The metrolink doesn't go all over the city, its pretty much a straight line, but it does have a stop at Busch Stadium, the Loop, forest park, union station (a former train station that's now a touristy mall), the riverfront, and other places like that.