From Singapore, the causeway take you into the heart of Johor Bahru city while the second link bypasses the city JB and takes you in into the toll/ PLUS expressway that passes through the length of the Malaysian peninsular and goes up to the Malaysia-Thai border.
However, some options yo may try is by sea. There are daily ferries going into Malaysia from Singapore into the state of Johor. Its a good option if you want to go to the eastern part of Johor, such as Desaru. ........more at From Singapore to Malaysia