I know of one place that usually has live traditional Irish music: John D McGurk's Irish Pub.

There is a free weekly tabloid called the Riverfront Times, similar to what you find in a lot of other cities, that contains an extensive arts calendar. There is also the daily paper, the St. Louis Post-Dispatch.

There is a good store that specializes in acoustic string instruments called Music Folk Inc. They also have a good selection of music books, but few CDs. There, you might be able to find out about house concerts and other events that don't get wide publicity.

If you like gardens, exotic plants, etc. don't miss the Missouri Botanical Gardens.

If you have kids with you, I recommend the Saint Louis Zoo, the Magic House (Children's Museum), the City Museum and Grant's Farm. (I think Grant's Farm is what Spaw had in mind when he said "Busch Gardens.") For cheap thrills, there is a McDonald's located in a riverboat (made to look like an old stern-wheeler steamboat) parked on the riverfront near the arch. They serve toasted ravioli there, too. For the price of a McDonald's meal or drink you can sit at your leisure and watch the barges go by.

By the way, the Zoo, the Art Museum, the Missouri Historical Society, and Grant's Farm are all free.

The museum beneath the base of the arch is called the Jefferson National Expansion Memorial. If I remember correctly, the museum part is free, but you have to pay to go up in the arch or to see the movies that are shown there.

And here's the St. Louis Convention & Visitors Commission site.