Hello Jasmine, my name is John, nice to meet you. To answer your question both are correct and mean the same thing. Natively in the US we use both interchangeably. I hope this helps.

Wait a moment - everydayish Wait for a moment - more standardized

Both are correct. The emphasis is different. "Wait a moment" puts the emphasis on the waiting. "Wait for a moment" puts the emphasis on the time period, a moment. "Wait a moment" is close to just plain "wait". "Wait for a moment" is close to "one moment, please".

I think "just a moment" is also useful

I also agree that “wait a moment” is correct.