what ways of communication on earth will also work effectively on the moon?

Answer 1

Depends on where (and how far) you wish to communicate.

Radio will be line-of-sight as there is no ionosphere around the moon to reflect radio waves beyond the horizon. You could use powerful radio and bounce the signal off Earth, as long as the other end of the communication is also on the 'Earth' side of the Moon.

You could place relay satellites around the Moon (but orbits around the Moon are tricker than around the Earth, because of Earth's greater tidal effect).

Microwave towers are repeaters. Expensive but doable.

Good ol' wire telephone and or telegraph would work. It would be fun to set up a railway line and use the rails as the conductors for a telegraph system. Then you'd really have a "way of communication from Earth that will also work on the Moon".

Semaphore (posts with arms or flags to relay simple messages from one post to the next -- line-of-sight, of course).

Answer 2

Radio.