Museum of Art and Archeology - On the University of Missouri-Columbia campus. A small but good art museum. A good permanent collection and lots of good temporary exhibits. They have lunch time lectures on art every week.
Bear Creek Trail - Columbia has great parks in general, but the Bear Creek trail has better scenery and fewer people than the MKT Nature and Fitness Trail. Park in the lot off N. Garth.
You can get there via the Missouri River, the Katy Trail, or even by a conventional road. Isn't that neat? See the website for driving directions, calendar of events, dog policy and more (it may just be my browser, but you may need to scroll around to see the info--it's worth it!).
Primitive camping, but with Wi-Fi.
The river valley is a few degrees cooler than the higher ground, so it's a nice place to avoid some of the opressive summer heat.
The live music is always good, and sounds to me like a blend of folk and rock. This year there have been some belly dance performances, and at least one fire-whirling show.
I love to watch the sun set over the river. There is always a nice bonfire. I'm not a night-owl, so I retire to bed before the end of the show. I like hearing it in the near distance as I fall asleep.
I love to eat a Cooper's breakfast while watching morning unfold over the river. There are some more-or-less constant residents there who meet in the mornings at a picnic table to engage in good conversation.
Relaxed atmosphere, and family friendly. I always see people with their young children.
Just south of the city on the Missouri River is a little gem of a place. They sell beer, snacks, BBQ, and sometimes Thai food. This business is located directly on the river and the Katy Trail (a converted railroad that is a bike trail that runs the width of Missouri). A great place to people and nature watch. You will find live music Friday through Sunday during the warm months. Camping is available.