Misshappy, welcome to the forum and to Singapore!
Your options for where to live are actually quite wide, given that there are school bus routes to most of the island. Obviously the travel time will vary, but it's not always 100% related to distance to the school as the crow flies - the exact route, number of pickup points and and your position along it will be factors too. To give you an idea, when we first moved to Singapore, we lived off Sunset Way (Clementi), which is quite close to TTS on the map, as we didn't want our girls to have a long school bus journey. That worked out fine, but interestingly when we moved further away (to Tanjong Rhu), their journey time didn't increase by any great amount. For a while we also lived in the Upper East Coast Road area and the journey was still OK for them.
So my suggestion would be to explore possible areas first and then check with the school, or directly with KAL Transport who run the bus service, to see how their routes work out for each location.
Of course if you plan on getting a car (personally I wouldn't), then forget everything I just said about the school buses.
As regards searching for properties and an estate agent, you might want to start by clicking on 'Housing in Singapore' -> 'Search Singapore
property' at the top of the page, as the owners of this site are in that business. I'd also suggest using the Search function directly above here and you'll find lots of discussions about housing, areas to live, etc. Then once you've had a chance to explore the options and available info, feel free to post any specific questions here and I'm sure the regulars will do their best to help.
Good luck!