You can take a bus from Woodlands MRT Station and Kranji MRT Station, take the bus to Woodlands Checkpoint, get off the bus and go through departure procedures, then take the bus across the Causeway to Johor Bahru and go through immigration procedures.
The Johor Bahru level is full of places selling stuff, nothing interesting.
Several other City Plaza and Jusco are also large shopping malls.
To the west along the coast is Danga Bay, a park.
There is a place in Johor Bahru where you can watch fireflies, but it is more troublesome to go there by yourself. If you want to go, you can sign up for a one-day trip to Johor Bahru from Singapore Travel Agency.
In general, if you don’t have friends in Johor Bahru and don’t have a car in Johor Bahru, don’t go there.
There is nothing interesting around the checkpoint, and you have to take a taxi to go further. The Malay driver’s English is very poor, and if you are not familiar with the place names, it is basically difficult to communicate.
I had been there before on my own and felt the pain of not knowing the Malay language. Later, friends from Johor Bahru came to the checkpoint to take me around.