I do the KL-Singapore via the Second Link very often, being a frequent visitor to Singapore.
About your trip to Malaysia this weekend :
After crossing the Second Link, follow the direction of Kuala Lumpur (do not go the direction of Johor Baru). You will pass through 2 toll booths before you hit the North-South Highway which will take you all the way to Kuala Lumpur. Estimated journey time : 3-3.5 hours.
If you don't go beyond 110km/h you need not fear. Otherwise watch out for speed traps. If you need to stop for a pee or a rest or whatever, there are several rest and eat places along the way. You won't miss the loo and the fork and spoon signages.
I assume that you will be travelling to KL on Friday but if it's on a Sunday evening, you will meet with quite heavy traffic as you approach Seremban, a town about 45 mins away from KL.
You will reach the Sungai Besi toll in about 3-3.5 hours. Make sure that you take the correct lane (there are about 15 lanes there) at the toll booth. Unless you have one of the 2 prepaid cards i.e Touch n Go or Smart Tag cards, please avoid the lanes reserved for them. These lanes have their blue and yellow coloured logo boards displayed at the top. Instead, you must take one of the TUNAI (i.e. cash) lanes. If you get into the wrong lane, it can be quite messy trying to wriggle out and into the correct lane.
After that toll booth you head into KL and much will depend on where you intend to go. I can't resist speculating that you will head for Genting, like most of my friends south of the border do haha.
Have a great weekend.