You can take the train from Woodlands, Singapore until KL Sentral in Malaysia.
I've traveled in such trains before and I can assure you that you will find the experience rewarding. Especially when your bunk/seat is next to the window.
Better start book. Tickets can get sold out quickly for weekend trips, and even weekdays during holiday season (December, February).
Click the link above. Do not be intimidated by the seeming complex train schedule. Basically choose your start point and end point and then decide from that range if you want a lot of stops or fewer stops (express).
Pros and cons: Lots of stops allow you to view the different landscapes and enjoy people-watching in every station that the train stops but it takes very very long time. Express trains have fewer stops but it gets you there faster.
And for comfort, please go for first class seats/bunks.
A one-way ticket should cost you around SGD20 plus. Can't remember the exact price.