K.E.
HI J., Welcome to Olympia!
My family has really enjoyed the kindermusik program with teacher Terri at Johanson's School of Ballet. They have programs for all age groups and for siblings too. I've met some really nice mom's there.
Also, the South Puget Sound Community College offers excellent parenting classes for infants and preschoolers. You can bring your children (there is a play room for them attached to the classroom) and it is very low key and offers really helpful information to become a better parent. I've met more great, people through this than anything else.
There is also a great baby gym, toddler gym, kinder gym program through the Olympia Center. Its inexpensive and really enriching for the kids and lots of time to chat for the mom's.
Once every other month there is Family Dance Night at Fusion. It is low cost, all ages, dancing fun. Many of the dances are led by Karen Kirsch who is a movement therapist. They provide popcorn, dress-up clothes, props for movement and some toys. I always feel really full and blessed after participating.
The YMCA both Downtown and Briggs have great playcare programs for the kids and all kinds of stuff for adults.
There are other more public venues that are family friendly and really wonderful like the Farmer's Market, Music in the Park, The Pet Parade, The Procession of the Species and Arts Walk.
Good Luck!