Go to the Crossroads, check out a few funky art galleries. Check out Grinders, a restaurant/bar with some great greasy food and hipsters lurking around. They also have some really decent concerts in the adjoining space.
Here is the concert schedule.
I'd check out the Westside too. There is a great new beer garden, appropriately named The Westside Local Beer Garden - that just opened that is a huge hit. They have some terrific garlic fries and a wonderful assortment of beers, and since the weather lately has been awesome, you won't even sweat as you enjoy the summer evening outdoors. Its a great place to people-watch and you may even make a new friend.
Read more about the Beer Garden
here and
here. The Crossroads is located just south of downtown, and you can reach it on the MAX. The Westside is just west of the Crossroads, easily walkable from the MAX.
A couple bars have fun trivia nights. I'd recommend
The Brick in the Crossroads for Trivia Riot. I believe their Trivia night is Friday night. I've heard Record Bar in Westport also has a fun trivia night.
The Kemper Museum of Contemporary Art has some more eclectic pieces than the Nelson Atkins Museum if you're into art and like more modern pieces. Its located just north of the Plaza.
I'll second the Mutual Music Foundation. I like Jardine's for a good jazz club too, located on the NE side of the Plaza.
If you like burlesque, there is a burgeoning burlesque scene here, I think they do shows in the West Bottoms. You can find out more
here. You'll probably want to take a bus or cab to the West Bottoms, although its not far at all from downtown.
I think I heard Bar Natasha re-opened? Can anyone confirm? I've heard they have an entertaining drag show, but I can't confirm that. They are located in the Crossroads.
Like seeing chicks beat each other up? There is a roller derby team that plays in KC - the Kansas City Roller Warriors.
Here is their website.