Can you drink a glass of lemonade on the moon with a straw? I don't think it's possible but want to be sure and the exact reason.
I don't think it is possible because the atmosphere on the moon is less than one hundred trillionth ($10^{−14} $) of Earth's atmospheric density at sea level. So the moons' air pressure would also be very (equally) small.
I suppose that somenone's power to suck an almost 'vacuum' to create an underpressure in the straw is not possible. 
Perhaps to increase the surface of your glass could help a bit.
I think that gravity doesn't have to do with it a lot because that is the same for the liquid in the straw and outside on the surface of the glass.
Perhaps I forgot something....
Because of big variations of temperature on the moon I suppose the liquid is not frozen nor evaporated.
And probably you should protect yourself and provide yourself with oxygen etc, but they are left out for simplicity of the problem.
https://en.wikipedia.org/wiki/Atmosphere_of_the_Moon
 A: Short answer: No.
2 reasons: You won't have any lemonade left, and you will be dead.
While the moon does technically have an atmosphere it is for all practical purposes a hard vacuum (see the second sentence of the Wikipedia reference). Any liquid in a vacuum will simultaneously boil and freeze, spraying out of the cup in the process.
You, standing in the same vacuum, will quickly expel the contents of your lungs and will no longer be able to suck on a straw. As you will lose consciousness almost immediately the cup (and straw) will fall from your hand and land beside your body. 
If you are in a spacesuit with a hole for the straw the pressure in the suit will promptly exit through the straw spraying any non-boiled/frozen lemonade everywhere. If the straw is on your mouth the air in your lungs will do the same thing - your lungs are not strong enough to resist the pressure difference.
If we rewrite the question to read "Can you drink a glass of lemonade in lunar gravity with a straw", meaning you do this inside your lunar lander, then the answer becomes "yes, no problem"
A: We breathe (and by extension suck) by changing the volume of our lungs and allowing the pressure difference generated by that action to move air into or out of our lungs. 
Boyle's law: "For a fixed amount of an ideal gas kept at a fixed temperature, pressure and volume are inversely proportional."
As we contract our diaphragm and raise our rib cage, the volume of our lungs increases (and the pressure inside our lungs decreases).Note that breathing in is an active process. (its a bit more complicated than that because there is an intermediary step where the pressure difference moves through the pleural space, but this is the basic idea)
To breathe out, we relax our diaphragm and allow our rib cage to fall. 
Here is a graphic that depicts the action of breathing and the changes in pressure and volume:

The biggest problem with sucking on the moon is that there isn't any atmospheric pressure to move air or whatever you're sucking into your lungs and thereby equalize the pressure difference. 
Another way of saying this is that there is no way that you can create a lower pressure within your lungs than there is on the moon. So air will always want to flow out of your lungs. 
So you are correct in assuming that the lack of atmospheric pressure on the moon would prevent you from sucking.
An example of this is if you stick your head outside of a car that is moving very fast. The air that is moving past you is at a (relatively) low pressure. if you are moving fast enough, you cannot breathe. 
A: I think the answer is a straight-forward No. You could not "suck" the lemonade up the straw. You would have to generate a pressure lower than that of the Moon's atmosphere, which you point out is many orders of magnitude lower than on Earth. So this is not humanly possible. Even if you could generate a better vacuum, the maximum height of liquid which this pressure difference could support would be microscopic. 
However, this does not rule out other methods of getting the lemonade out of an upright bottle which do not rely on atmospheric pressure. If you used a very narrow straw you could draw water up by capillary action - ie adhesion between the lemonade and the material used to make the straw. You could also siphon the lemonade out of the bottle, since this depends on gravity not air pressure. But you would need to get the siphon started, and probably need to ensure the lemonade is de-gassed. See Do siphons really require air pressure to function? and The height limit of a siphon.
