if u want to be near the greenery for walks and yet be in the city (near malls), try the Tanglin/Grange Road area as it's near the Botanical Gardens.
Or live out on the east coast where there are accommodations stretching from Tg Rhu/Fort Rd/Meyer Rd area (nearest to the city) to Bedok (nearer to the airport).
I wld highly recommend Siglap area (like a village) and if u choose a strategic location to stay, u are walking distance to the beach.
In terms of childcare, there are plenty of choices: live-in maids, nannies, childcare centre.
If u r not working, there are some playgroups (mums support group) esp. for people like u - settling in to singapore with kids. U'll get to meet many other expat families here.
Other than playgrps, there are the established franchise such as Gymboree, Kindermusik to bring your toddler for activities.
Otherwise, depending on where u r coming from, it's having summer all year round.
good luck.