"On the bus" is a fixed conventional expression in all varieties of English across America and Britain. "On the bus" is used in the transportation context (i.e., you are on the bus when you are travelling by bus) while "in the bus" can be used to show your position (in the bus). A similar example is "On the Train". You say "on the train" when travelling by train whereas you use "in the train" to indicate your position in relation to the train.
However, the reverse is the case when it comes to cars. You are "in the car" when travelling by car. In this case, it is inappropriate to use "on the car" except if you are on top of the car. It is also a fixed conventional expression.
You can use the hints below to get them right:
1. You get OUT of a car, so you get IN it.
2. You get OFF a train, so you get ON it.
3. You get OFF a bus, so you get ON it.
IN for OUT, and ON for OFF. It is just like using the opposite of one to get the other right.