Another way of looking at it is whether the vehicle is public transport (taxis count as private hire). So "get on" for buses, trains, planes, boats and so on, and "get in" for cars and taxis. If you are physically on top of the method of transport, then of course you use on: get on horses, bikes, motorcycles, elephants...

There isn't a good answer for this, it is just one of the little features of English. A good rule of thumb you can use is: If it is large enough to stand up in (A bus, A boat, an airplane) say "get on". If it is smaller and you must always sit (A taxi/car) say "get in". That won't be correct 100% of the time, but should help most of the time.

Looks like you got 2 good answers below Claudia. English is a funny language!