In Malaysia there are a lot of bus connections available as well, which you also find via the booking links.
If you want to travel from Malaysia (arriving by train) to Singapore, you have to use first a bus to cross the border and then the Singapore Metro.
You arrive at JB Sentral (the station of Johor Bahru, Malaysia). There you change to the bus number 950 in direction of Woodlands MRT station. This bus cross the border (border control) and its journey time is about 45 minutes.
Change at Woodlands MRT station to MRT Singapore Metro and continue to a station in the city center of Singapore. Journey time as well about 45 minutes.
Metro and bus run in frequent, short intervals. Total costs are less than EUR 10.
If you want to travel from Malaysia to Thailand by train, buy your train tickets via the given booking links or at a station.
The common travel route from Kuala Lumpur (Malaysia) to Bangkok (Thailand) is as follows:
By train from Kuala Lumpur to Butterworth in the morning. In Butterworth you change the train for Padang Besar. Padang Besar is the border station between Malaysia and Thailand.
In the evening a direct night train starts from Padang Besar (Malaysia) in direction of Bangkok (Thailand).
For every single trip of this journey, you have to buy a separate ticket via the booking links.