I'm 38 and a mother of an almost 2 year old. It's funny how it doesn't matter how old you are when you have a kid...it's hard to find stay at home mom's to hang out with.
I don't know about any playgroups, but as for activities, here's a few:
Deanna Rose Farmstead at 35th and Switzer is a free petting zoo/playground/trails etc...We love it. It's big and there's alot for little ones to run around and do.
During the School year, the Roger T. Sermon Center in Independence MO has an open play time for preschoolers from about 10-2. You can search their website and get the details. I think it was Mon., Wed., and Fridays.
The Nelson Art Gallery has free admission. They charge to park, but you can park on the west side of the gallery on Oak St. for free. Plus they have a nice huge lawn with more art sculptures on it.
Parks: At about 31st and Gillham, there is a great park that has some really cool stuff to play on. There's a huge pirate ship, fiber glass cars, fire engine and stuff.
There are a couple of sprayground parks in Independence. One is at McCoy Park at 24 hwy and Deleware and the other is at Rotary Park at 24th and Hardy.
Burr Oak Woods is in Blue Springs. They have an indoor nature museum and outside there are several trails.
Kaliedescope is around there too. It is for kids 5 and over...and it's free. It's run by Hallmark. You can find it online too to find out when the sessions are. I always took my 4 year old...what are they gonna do? Check ID?
Hope this helps you out.
K.