The short answer is a simple "yes." If we are going on a trip, my wife will say to me "we'd better buy our plane tickets soon." The long answer: on trains in the United States, it is simple. You buy the ticket, you show the ticket to the conductor, you board the train. In plane travel in the United States, there are two different things: a) a "plane ticket" and b) a "boarding pass." The ticket proves that you paid your money and have a seat on a particular flight, but the boarding pass is the document you need in order to get on the plane. The boarding pass is a printed card you show to the gate agent. The plane ticket might be a printed card, but increasingly these days it is an "e-ticket," a purely electronic transaction.

For a train, yes. For a plane - historically, you did buy a ticket. But now, mostly people buy online. Then, when you go to the airport, they give you a 'boarding pass', which is what you show when you actually board the plane.

Yes, but I am confused by your use of 'also' - Also what? when we refer to a ticket which will allow us to board a plane for a flight somewhere it is called a plane ticket (Amercans might say airline ticket? - not sure about that). And a ticket to board a train is a train ticket. For buses, it is, of course, a bus ticket.