If you’re into amusement parks, Worlds of Fun is still open for the season, although only for the weekends this late in the year. (They have a haunted special attraction going on now, I believe.)
The haunted houses are open now. The Edge of Hell is a popular attraction, as is The Beast (owned by the same folks).
As others have said, The Nelson-Atkins is a great museum, and their Asian collection is top-knotch. If you like modern art and want to see more than what’s at the Nelson, you might try the Kemper Museum of Contemporary Art.
The Plaza is, as others have mentioned, a great place to shop and eat. It’s down the road from the Nelson, too, so you can visit the museum and grab something tasty without having to drive all over town.
If you’re into theater, The American Heartland is usually a pretty sure bet for a good performance. The Unicorn, is too, although they tend to put on a lot more angsty plays. It’s a smaller, more intimate setting, though, so for good or bad, you get a closer view of the actors performing.
The majority of the bars and clubs in Westport are closer to the intersection of Broadway and Westport Road (so, if you’re in to clubbing, that’s the area you want to focus on), but Westport extends beyond that in both directions. It’s fun to walk around; some of the smaller shops are college-crowd, anti-establishment hoots.
Definitely be sure to try some KC BBQ while you’re here. Some of the best in the world! My personal favorite is Jack Stack (one location on the Plaza), but there are plenty of others. Danny Edwards ran Lil’ Jake’s Eat It and Beat It, but it was recently taken over by a construction project, so he’s opening up new digs elsewhere in town. I’m hoping it’s open by the time you get here, so look for it when you get into town.
I second the The Liberty Memorial, and the gorgeous Union Station is right down the street.
We also have four major casinos in town, with Harrah’s and Ameristar being the largest.
If you want a comedy club, we have Stanford’s.