On a plane is correct. On refers to being on a surface, and a plane and other things such as buses and ships have "platforms" that are big enough to stand on.
However, you can also say in a plane or bus, because in means enclosed in.
Similarly, you can say I'm in or on an elevator, depending on whether you conceive of yourself standing on the surface (platform or floor) of the elevator car or of being inside the elevator car. Of course to stress that you are on top of a car, plane, bus, elevator car etc. you would use on top of.
As for chair, you can either be on a chair, which means in contact with its (horizontal) surface, and this could mean the seat, the arms or the back of the chair; or you could be in a chair which means enclosed by the chair.
For an extended discussion, see
"In" and "on": How can I decide which one to use for vehicles?
and the related
Q: Should I say “She is in the park” or “She is at the park”?.