I lived in Kansas City (and environs) for the last 26 years, until about two months ago.
You will need a car to do most things. Period.
Kansas City, KS is tiny compared to Kansas City, MO. Everything I mention here is in MO.
You really want to try in the Country Club Plaza/Westport/Union Hill area if you want something to do every day. The Plaza is upscale shopping and dining. Westport is lower-scale funkier shopping, lots of live music and food. Union Hill is a pretty place to live.
Downtown (google the City Market or River Market) is nice, but can be expensive). It’s been revitalized in the last couple of years.
The Nelson-Atkins museum (on Oak near the Plaza) is great, don’t miss the shuttlecocks on the lawn. Across Oak from the Nelson is the Kemper Museum, which is often better than the Nelson, not as “museum-y”. They have revolving exhibits. Say hi to the security guard near the entrance to the restaurant, which is expensive, but beautiful.
If you like teenagers, hit a haunted house under the 12th St Bridge (yes, that’s the address). They’re in these awesome old abandoned buildings and the parking and entrances are under the bridge that 12th st forms over the west bottoms.
If you do have a car… Powell Gardens is in the far eastern suburbs and quite nice. Burr Oak Woods in Blue Springs is nice for trail walking, as is Corporate Woods, but bring pepper spray if you plan on walking alone in Corporate Woods there was a rash of attacks 10 years ago or so.
On the Kansas side:
If you do have a car, go to Lawrence for a day (30 miles west). Home of the University of Kansas, has a great downtown (Massachusetts St). Eat at the Free State Brewery. Have coffee at La Prima Tazza or Z’s Espresso (or any of the other 10 coffee shops downtown). Avoid Coldstone and eat ice cream at Sylas and Maddy’s. Drive around Lawrence, it’s beautiful. If you go on a weekend, you can drive on campus.
If you’re in Overland Park or Olathe (oh-LATHE-uh), there’s not a lot of walking stuff you can do, like you can in KC MO proper.
Weather:
It’s completely unpredictable. Seriously. Get a jacket (like a fleece pullover) and wear layers every day. It probably won’t get really cold, but I have seen snow in early October. It will be cold in the mornings (40s) and warm (up to 80s) during the day. Or, it could not get above 50. You can’t tell.
Have fun!