A couple of friends of mine rent cars - long term lease. I think it goes from about 1.5K per month upwards - google lease/hire cars Singapore for more info.
We've never bothered with a car here - been here 10 years - our children were 3 and 5 when we moved; they adapted to bus travel quickly. They love it when we go abroad and hire a car though - it's a novelty.
Housing - I'd honesty wait until you've started the enrolling in school process before thinking about areas. It is nice to live quite close to the school, especially when your children are small and it seems like you're always up at the school to watch an assembly or some performance (parents are not usually allowed to get the school bus with the children).
Many children get school buses to school (even tiny children) - they have 'bus aunties' who keep an eye on them. However, school buses can be costly especially if you go for international school (local school buses are cheaper........just another way to rip off the foreigners - it's not like fuel costs more for international school buses than it does for local ones

).
Singapore is easier for a Brit expat family than some - mainly those for whom English is not their native language. I've got friends from non-English speaking countries - they've all had to brush up on their English since living here.
Serviced apartments are a good option for those newly arrived. You usually have to have your employment pass before embarking on things like renting apartments, opening bank accounts etc (maybe even enrolling in school.........it has been a while since we enrolled in school but I'm sure we had to have the passes first).
Search the archives - there's always loads of stuff for those new to Singapore (I think you have to be logged in to search though). The site mods have got some info for newly arrived expats too if you search the pages other than the forums.