Taking the bus from Johor Bahru, Malaysia to Singapore is easy once you know what to do.
First go to the Custom and Immigration Quarantine Complex (CIQ) in downtown Johor Bahru right near the causeway to Singapore. Walk through the complex to leave Malaysia (you need your passport, obviously). Once you clear Malaysian passport control you follow the signs to Singapore/Woodlands for the bus (on your left after you leave the passport area). You go downstairs and catch your bus. You pay for the bus, on the bus.
There are various options but the 3 most common are:
The busses will take you across the causeway to Singapore and then you will disembark (with any luggage and belongings) and go through Singapore customs and then go back downstairs and catch your bus. The CW1, CW2, 170 and 160 run frequently during the day and evening (every few minutes to every 15 minutes for some).
Overall it is an easy process and doesn’t take too long. But at rush hours (especially Friday night or on holidays) it can get very busy and backed up. I hear the morning rush from JB into Singapore is pretty bad, but I have gone a bit later (9:30 AM) when I have gone, and it has been fine (most of the time).
The prices are from Singapore to JB are the same number but in SGD which means about 2.5 times as much money. So, for example, from Queen Street to JB is $3.30 (versus 3.30 RM).
It is also possible to catch the CW3 and CW4 over near the 2nd link and take that into Singapore. For more details on the CW buses (yellow) see the Causeway Link web site.
When going to Johor Bahru from Singapore you can exit the CIQ after passing through customs (to go in downtown JB – City Square…) or take the bus to Larkin Bus station (and I think some also go on to other destinations in Johor Bahru).
Taxis are a bit of a pain between Johor Bahru and Singapore. Normal taxis can’t go between the countries. There are certain taxis that are allowed to. From Johor Bahru they will take you to Queen Street in Singapore (that is the only location you can go to). And from Singapore you have to catch it from Queen Street and it takes you to the downtown bus terminal (near Landmark Mall – a dying mall). From Singapore to the Larkin Bus terminal in Johor Bahru is SGD 40. You can take the taxi yourself and it costs that. Or you can split it with others waiting (up to 4 others so $10 each). If you want to go to somewhere else in Johor Bahru you can but it will cost $50 (or more). There are other special taxis that can go to the airport and maybe other locations (they are expensive).