A.H.
Do you belong to any playgroups? They usually are free, or charge VERY minimal fees (like $12 / year) and have playdates at homes, parks, etc. Try looking for one in your area at www.meetup.com
The Montclair Art Museum is free every 1st Friday from 1-5. The Bronx Zoo if free every 1st Wednesday (I think - you should check their website).
IKEA and Jersey Gardens have free play areas for kids.
Oh, Yes! Thanks Beth C! Check the libraries. The Bloomfield one is great. Other libraries (Nutley, and Clifton for sure) have storytimes, and the Barnes & Nobles in Clifton (Rt3) has storytimes too.