East Coast is generally considered a good area for people who want a bit more peace and quiet. It's a bit more spread out and somewhat lower-rise. Then there's proximity to the beach and facilities, the boardwalk trail and so on. It's popular with families.
On the north-east side is Pulau Ubin [island] which might be one of last semi-wilderness parts of Singapore. You can go and hike there, depends how much you enjoy the punishing climate.
There are also some forest/reserve areas that serve as the national water catchment-area, like McRitchie Forest and Reservoir which has several hiking trails, that is located in the central/north area.
But SG in such a small place that you don't have to live on the doorstep of such places to enjoy them on the weekends, though it would make access a bit easier. I think most people work long weekday hours, so access to open space matters less on work-days.
Have a look at the 'Where to Live' FAQ, and it might spark some advance thinking about which areas could work best overall for you, with regard to workplace, amenity, commute time, your budget etc...>
http://forum.singaporeexpats.com/viewtopic.php?t=101985
ps. Malaysia's Tioman Island is a c30 minute flight away and a world apart. Unspoilt, pristine beaches, some great hiking. You'll find plenty of divers in SG visiting there for as many w/e's as possible during the season.
Diving off Singapore doesn't fit the stereotype of a tropical idyll. Not least as SGs harbour is one of the biggest in the world, the water is turbid and there's not much left of any coral. So it's more a kind of macro or muck-diving style there. Fun for a few dives, but you don't relo to Asia to experience diving in like a warm-water version of Brighton England
