Hi Dolphin,
Another interesting place & free place which is accessible by public transport (aka bus) is the fire station. Not sure how old your kid is but my friends and my kids enjoyed it thoroughly when we visited the Waterloo fire station a few years back. We get to see the fire men doing drills and get wet ourselves with the fire hoses.
I would recommend you to visit the firestation at Sengkang, supposedly newest and biggest in Singapore I think. And best of all, easily accessible via bus # 82 from Punggol (outside Punggol Bus Interchange...not inside).
All SCDF's fire stations, except Jurong Island Fire Station, offer open house on every Saturday from 9.00 am to 11.00 am. We didn't do any booking for the visit, we just walk in the last time. But I believe if you have a big group, some booking is required. No harm calling up to check before your visit if you like.
http://www.scdf.gov.sg/Community_and_CD_Volunteers/E_Services/fire_station_visit_faq.htmlWhen other interesting and cheap places for children comes to my mind, I will share with you again. Good luck and hv fun with your kid.