Get Out Magazine appears every Thursday in the Post-Dispatch and lists lots of events too. Places to go and things to do The Missouri Botanical Garden has already been mentioned. The Arch is actually pretty impressive, especially up close. Note that it's usually a long wait followed by a long ride to get to the top, and the view isn't that impressive. Perhaps the most interesting thing about going up is the bizarre elevators themselves.
Most folks find the museum more worthwhile.
The City Museum is definitely worth a visit. The St. Louis Science Center is pretty cool, too. The St. Louis Symphony Orchestra has a rich tradition and Powell Hall is tough to beat as a performance space. The St. Louis Cardinals have an equally rich tradition, though I can't say they're as consistently brilliant as the orchestra. The International Bowling Museum and Hall of Fame is another one of our local attractions.