We are under Liverpool Council and the only one I see targeted on their website are childcare services. http://www.liverpool.nsw.gov.au/services/childcare-centres. There are also some meet ups at liverpool library but not sure when it runs anymore. They were usually reading groups for under 5's. Other than that, please don't illegally dump near Cabramatta Creek. They offer free 2x council pick ups for your old furniture/white goods etc.
If you want specific activities, the local PCYC have pretty good facilities and after school programs (small fee). The Hub at next to Miller shops have some health counseling to offer.
My brother goes to Ashcroft High and it has been given a large amount of funding due to its low socioeconomic status but also due to the high aboriginal enrollments in the school. One of them includes free breakfast or cheap lunches and some allied health services free of charge for referred students. I guess the place to start if you were to inquire about additional activities is the school.