we also do the rounds of the parks in good weather. There is a big park down by the coast - marina city park. you can get a bus to it from Chinatown. There's a play area there, but there are also statues and sculptures which are quite interesting. it's very peaceful there as it is a bit out of the way - we managed to buy drinks from the food centre nearby but it would be worth taking your own just in case it isn't open.
Other (free) places to go apart from ones mentioned here are Chinese/Japanese gardens, Botanic gardens, Fort Canning Park, East Coast recreation centre (small play area outside McDonalds), or HDB estates which always have play areas. McRichie reservoir is quite interesting for kids, and you don't have to walk all around it, you could just stick to the garden area and watch the monkeys and monitor lizards. you can also have a cheap day out at Sentosa, just pay the entry fee and walk around the gardens and go to the beach.
You can also join the zoo for around $80 for a year and then you get free entry. I don't know where you are from, but where I'm from $80 is about what we'd pay for one visit to a zoo back home, so it was worth doing for us. I think the bird park has a similar scheme.
Tampines mall has an Ezone place with a play maze in it, which might not be suitable for a toddler, but something to bear in mind when they get a bit older.
