Why we are able to see the moon?
We are able to see the moon because the sun's light reflects off of it. We are able to see only the part that is reflecting the light.
As long as Earth is above the horizon - and this happens if the astronauts are on the side of the Moon that faces Earth - they will always be able to see Earth. Earth would look four times as large (in diameter) than the Moon looks from us, so about 2 degrees; also, much brighter than the Moon looks to us, because of the larger surface area, combined with a significantly larger albedo.
The moon goes through "phases". Sometimes the moon is opposite the sun, other times it's on the same side of the earth as the sun. During THAT time you won't be able to see the moon at night. You will also not be able to see it if it has not yet risen or if it has already set. Some believe that the moon is always in the night sky. That is not the case.
Well, not right now, because the Moon is only three days away from the New Moon, so it is quite close to the Sun in the sky. But if I get up about 2 hours before dawn tomorrow morning, I will be able to see the waning crescent Moon low in the eastern sky. And if we look very closely, we may be able to see the "dark" part of the Moon, illuminated by reflected…
"New moon", as used in Astronomy, is technically the "no-moon" - when the Moon is closest (in direction) to the Sun, so that you can probably not see it. When you see a crescent, perhaps 1 or 2 days after the new moon, the moon will set soon after. At most, you should be able to see the moon for 1-2 hours - depending how much leeway you leave in your definition of "crescent".