The "hill" area is fantastic if you like italian and not too far but you will have to cab it. There is a "Charlie Gito's" location downtown though (just a few blocks from the stadium). It's a famous italian establishment with a lot of photos of celeberties on the walls such as Tommy Lasorda who frequent the place. Charlie makes lasagna to die for. Link below:
http://www.charliegittosdowntown.com/
You could head down to the landing by the arch which is walkable (or short cab ride). There are a number of establishments there including a couple of nice brew pubs and a lot of action. See the original bridge accross the mississippi while down there too. If you want you can jump on a metro there and go accross the river to the casino on the Illinois side.
Mike Shannon's is a nice place and I believe there is still a Dierdorf & Hart's downtown still if you want a upscale dinner (steaks, etc).
There is also ballpark village which is adjacent to Busch Stadium. Theres a number of bars, clubs, and eateries in there. The ratio of women to men at the clubs in the evening is very appealing there...wink, wink, nudge, nudge.
Theres also Union Station which you can take a short metro ride to (or possibly walk but its several blocks) where you'll find a Hard Rock and a Landry's seafood resturant and others including lunch type joints (ocharley's, etc). I went there the other day and wow was it dead though. I wonder if ballpark village is hurting them that much? It's worth seeing as a landmark anyway if you have the time.
The Loop area is another one that has some cool places to eat including Fitz Rootbeer and Blueberry Hill but again you'll have to either cab it or take the metro.
The Schlafly Tap Room is one of my all time favorite places. Great beer made on premises and unique food selections. You will probably have to cab it but its not that far. It's pretty close to union station. Link below:
http://schlafly.com/tap-room/