the botanic gardens is a great place to walk around in.
try suntec city at the koi gardens, where your child can feed the fish, if you go before 11 am (the petshop is still closed, so bring large pellet koi food). there is also the fountain of wealth also in suntec. if you have spare money, try the duck tours too.
my 3 year old has always loved water, so we try to rotate visiting the floor fountains at the following areas: bugis junction (bugis mrt), united square (novena mrt) and the water area at the zoo. bring extra clothes for sure for yourself too.
we also frequent the public library.
hope this helps, i know it feels confining just to be at home. goodluck!