Either one is perfectly fine, Namlan. The reason I asked why you think "drives" is better is because your first sentence ("takes me") is very idiomatic. In this case, I really don't think that either sentence would be better than the other.
The other thing is that "takes me" is explained by the reference to the vehicle. We know, without having to say so that you went on his motorbike because you've said so.
And yes, you could say "takes me to school in his helicopter" or "takes me to school on his sports bike". This is a completely natural phrase.
You would also hear:
"I'm taking my little sister to school on my bike" or "I'm taking my granny to do her grocery shopping" or "I'm taking my Mom to the hairdresser". "Taking" used in this context is perfectly natural.