As a start, i'd recommend your first few rides out with a cycling group. There are plenty around and if you Google them, Facebook pages pop up. Go with the ones who respond to you but make sure the group bikes at your comfortable speed. Some of them go for longer distances than others and maintain different speeds so you may need to be at a certain fitness level to keep up.
Once you're comfortable cycling on the road with the group, you can have a go on your own. There aren't bike lanes on the roads and you need to keep the far left lane at all times. It can feel a little stressful when you've got a bus trailing behind you because there's a bus stop 100m ahead or they can move into the right lane to overtake you.
Overall, more and more people are hitting the roads, especially on weekend mornings. Humidity is a serious problem so most people try to end their rides by 8am (9am latest unless you have a desire for heat stroke). It's also safer to cycle in the wee hours of the morning as there are fewer cars on the road.
Drivers in Singapore aren't exactly the best and there have been hit and run cases reported. This isn't to scare you but to warn you so you stay alert. Sometimes, it doesn't matter how many lights you have attached to your body, if the driver is terrible it's a matter of luck.
FYI, road bikes aren't allowed on buses and trains. If you get a flat tire along the way, you can expect to spend a long time trying to flag a cab down as drivers are reluctant to stop for cyclists because loading the bike into the cab is a hassle.
