If we looked back at Earth from the moon, what phase would you see when the moon was full? new? a first-quarter moon? A waxing crescent?

If the moon was full, you'd see a "New Earth".
If it was New Moon, you'd see "Full Earth"
If was first quarter, you'd see the 'half Earth', (at 3rd quarter)
If the moon were a waxing crescent, you'd see Earth's waning gibbous.

New. It's obvious if you think about it - at Full Moon, the Moon is directly opposite the Sun in the sky so the whole of it is lit up. What you would see of the Earth is the night side.

New.

If teh MOON was full, the Earth would be "new"