The answer to this is lost to prehistory.
The first to look at the moon through a telescope would likely have been its inventors: Hans Lippershey, Zacharias Janssen and Jacob Metius, in 1608.
We see the full moon {the phase}.
The moon.
After a week from new moon, you'll see a half moon. This is more correctly called the First Quarter phase of the Moon.
on a first quarter what do you see
You see the shadow of Earth cast upon the moon. The Sun's light creates the shadow and illuminates the quarter moon you see.
If you are referring to the first moon landing by Apollo 11, you see Neil Armstrong first followed by Buzz Aldrin.
I'd have to say the first scientist who saw the moon was the first one to look up in the sky...
When the right half of the moon is illuminated it would be called first quarter. (That's when you see the Moon from the northern hemisphere.)
His eyes.
Italian astronomer Giovanni Battista Riccioli was the first to see and document mountains on the moon using a telescope in the 17th century.
Galilao mapped the moon and saw jupiters moona
You often CAN see the "dark side of the Moon". It's easiest to do around the first quarter Moon or a day or two before that. The Sun shining on the Moon causes the Moon to reflect light, and we call this "moonlight". But if you were on the Moon and looking at the Earth, you could see the Earth by reflected sunlight, which we would call "earthlight". When we look at the first quarter Moon, we can often see the dark side of the Moon illuminated very slightly by reflected earthlight!