on I-70, you probably need to drive. There is the MetroLink, but I'm not sure that it goes from Harrah's to Busch Stadium. It does go from Lambert Airport to Busch, though. So, you could maybe take a shuttle from Harrah's to the airport and get on it there.
Like I said, it's probably better to drive. Traffic on gameday can be pretty hectic. There are a ton of parking garages downtown near the stadium that are within easy walking distance of Busch. I think it costs about 15 dollars to park. I think you take the Memorial Dr./Market St. exit- I usually am coming from I-55 N. and I take the Memorial Dr. exit from that side. It's not real hard to get there because you have so many landmarks.
What I do when I go is stay at a Hotel near the stadium- there are several very nice ones near the stadium- and just walk. I stay at the Hilton by the Ballpark- it's connected to Mike Shannon's Restaurant and it's within walking distance of the Arch, LeClede's Landing, Union Station, and the Courthouse. It's pretty nice to just walk to the game because you don't have to worry about parking, the traffic or any of that other stuff. I will say that the traffic situation seemed to be a little bit better this year as opposed to the past. If you drive, you might want to do something like go to the Arch or the Landing until traffic clears out a little bit.
On the way up, Lambert's in Sikeston, Mo. is really good. It's like a Cracker Barrel but 100x more awesome.
Also, there is/was a St. Louis Cardinals museum, but I think they've moved it, and I didn't see where or if it was open. It used to be in the bowling museum. But it was really good.
Also- The Anheuser Busch Brewery Tour is very good. The Hill is the Italian part of town, and there are some very good restaurants there. That's where Yogi Berra and Joe Garagiola grew up, and I think Jack Buck used to live there at one time.