B.T.
If you're in the North Kansas City School District, you should go to their website and look up the Parents as Teachers (PAT) Program. Among other great activities and benefits (funded through the school district so no additional cost to you besides paying your taxes), they have a playgroup program where moms & their kids meet twice a month at the early education building in Pleasant Valley. The room has rocking chairs for moms and tons of toys & books with which the kids can play.
The mid-continent library system in the Northland also have quite a few activities for young children as well as weekly storytimes usually followed by a craft activity.
And there's a group called mommy & me that meets at a church somewhere on Vivion Rd. They have tons of social activities and opportunities for playgroups. I don't know where they hold playgroups though. I think the cost of membership is pretty low. I'm pretty sure there is a mommy & me website to get more info.
The Zona Rosa Shopping Center has a small indoor play area where the food court used to be "under" the Marshall's store. And some moms go to the Metro North Shopping Mall and let their kids run around the "blue carpeted area" near the big water fountain with the hot air balloons that go up & down. It is located on the first level. It's best to go through JCPenney to get there. The mall is losing a lot of stores and many of their "snack & food" places but it's better than nothing.
I have a 2.5 yr old son and 6 year old daughter. I've been in KC for almost 9 years. Maybe we'll see ya around town.