Well, if you’re into vintage clothing and fun stores, and since you’re going to see Radiohead I’m going to assume we’ve got similar tastes. I am too poor to see Radiohead, although I’m sure it is going to be well worth your money… Anyways, here’s a few things to do:
Downtown: Arch (obviously), and the riverfront is pretty nice. There are a lot of bars downtown, but I think they’re all pretty crappy. You can always eat at ‘The Speghetti Factory’ which seems to be high on the tourists ‘to-do’ list, but I don’t think it’s really great. There’s also the Budweiser Brewery that you can tour, it’s a couple minutes from downtown. I haven’t been there for a while, but you do get free beer I think.
South Grand: This is a good place to be if you want to check out some vintage clothing stores and some good little places to eat. It’s a little strip on Grand Ave. To get there, you’ll take hwy 64/40 West from downtown and take the Grand exit (and head south). Or, if you’re in the west part of downtown, you can just hit Grand from some of the streets there. I really would suggest eating at ‘The King and I’ if you enjoy Thai food. It’s a nice little place, and I THINK that they’re open for lunch, but don’t know for sure. Right by the restaurant, there are a few stores that you might enjoy.
Central West End: This is an ok place to go, there are again a few stores of interest and a few places to eat. Not too bad all in all. To get here, you’re going to head west on hwy 64/40 and take the Kingshighway North Exit. There are some pretty funky stores around here, but the streets are nice to walk down. Just make sure you don’t stray from the shops too much, it’s a rough world out there.
University City: I like U-City, but I’ve only been there during the day a few times. This is an area that is really getting a nice (and much needed) facelift. You can find some interesting shops there, and all kinds of fun things. There are some good restaurants, and good music/book/clothing stores to visit. Plus, while you’re there, you can get a tattoo.
Kirkwood: This is for the Yuppie in you. You may want to stop by, but There isn’t a whole lot to see. It is pretty though.
Forest Park: This is nice if you feel like walking around, or seeing the zoo. We’ve been having penguins escape from the new puffin/penguin exhibit, so if you’re lucky, you’ll get hit by a penguin. The zoo is nice, and free, and big. The rest of forest park has some nice stuff to look at, but not a lot to do other than walk or golf.
There are a few other things to do in STL, you could visit Grant’s Farm which is nice, there is the Shafly Brewery (which is really close to my house), the Limp Mansion which is haunted (you can eat there and take a tour I think), there’s a few malls, but you can see those anywhere.
Well, hope this helps you out, and now that I’ve done this for you, I’m going to Memphis (well Tunica actually) in October, so sometime I need a guide to your city! 
Here are some links that will at least give you an idea of where places are, and you can map out your plan of attack.
Here is a poor overview of the city:
http://www.bradleyanddana.com/images/stlmap.jpg
Here are some links to places in neighborhoods I mentioned. Directions to any of those places will get you to the neighborhood.
South Grand (The King and I): King & I in Saint Louis, MO 63118 | Citysearch
U City (Blueberry Hill):
Central West End (The Grind): http://stlouis.citysearch.com/profile/5736860/
Forest Park (The zoo):
Kirkwood (Humble Abode Furniture):
http://stlouis.citysearch.com/profile/5749330/
Downtown (The Arch):
Budweiser:
Hope this helps, and let me know if you have any questions! I made these recommendations based on the fact that you like radiohead and were looking for something different. There are a lot of places that are more ‘tourist oriented’, but these places are pretty neat.
-Brad