Learn Russian: Лифт едет вверх.

In Russian, you ride in an elevator, so it is treated like a vehicle. While in English an elevator just goes, Russian makes a strict distinction between going on foot (идти) and going by transport (ехать). Because you are transported by the elevator, it uses едет.

Why is it едет and not ездит here?

The verb едет is a one-way (unidirectional) verb of motion. This sentence describes an action happening right now: the elevator is currently on its way up in one direction. If you wanted to say the elevator travels up and down all day (a repeated or general habit), you would use the multidirectional verb ездит.

Could I use наверх instead of вверх?

Yes, they are very similar and often used interchangeably. The small difference is that вверх focuses purely on the upward direction in space, while наверх usually implies a specific destination, like moving to the top floor of a building.

Is лифт related to the English word lift?

Exactly! It is a direct loanword from the British English word for elevator. Because it ends in a hard consonant, it acts as a completely regular masculine noun in Russian.