Good Morning - well at least here...
Shooting the Moon, I don't think is dependent on a single item - such as a lens. But, I can certainly understand what you are asking. 300mm is a pretty long lens (or a zoom out to 300mm), and should do the trick. As such, it is not going to fill the frame, so you are going to need to crop. All of that said, there are a whole host of other items to consider and address, that goes into getting a good shot.
The Moon is a bright spot usually in a dark sky (well at least at night). You camera has at least two main exposure approaches. One set averages the light over the entire frame (or a good part of it - such as center averaging). The problem here is that the Moon is a very bright spot and everything else is pretty black. So, these modes really do not help. There is also spot mode, which uses the point at the center of frame to meter off of. With the Moon being so bright, this is the mode that you want to use. [Note - after shooting the moon, reset the camera back to the mode it was previously in - or you will wind up with the camera acting different than before.]
With the Moon being so bright, you can use ISO 100 or 200. You can also stop down to f8, or so to get to the sweet spot in your lens. Also, you can go completely manual and use the "Sunny 16" rule - f16, what ever ISO speed, with the shutter speed the inverse of the ISO - which is f16, ISO 100 @ 1/100 sec; f16, ISO 200 @ 1/200 sec, etc.
Now, the Moon moves - pretty rapidly through the sky at night. Also, it is bright enough that you really don't need a tripod - you can shoot hand held, since your exposure is going to be pretty fast. But that does leave one aspect that you need to address - focusing.
You are going to have either autofocus or manual focus lens. If its autofocus then point it and get it (the moon in the center of the frame) so that the camera will meter on the light and lock focus on it, then shoot. With a manual focus, you will be trying to manual focus on a moving target while you are bobbing around (handheld camera) in the dark. What you might want to do is to go out while its still light and pre-focus on something that is far far, far away, and then don't touch the focus again (perhaps putting a piece of tape across the focus ring).
Then you need to shoot - if your hand holding, get in to a stable position, breath in, breath out while aiming, and when you finish exhaling, push the shutter.
You can get a doubler for your lens, a 300 lens with a doubler goes to 600mm. You get a larger moon, fills more of the frame, but its harder to hold and aim, plus as you are holding it, you are bobbing around more.
Also, it takes some practice.
If you are using a tripod, since the moon moves, you are always trying to readjust the aiming on the tripod to keep the moon in the frame. With using a tripod it helps to have a remote shutter release, so you don't need to touch the camera (shaking it) to take the picture. Or you can use the 2 second delay.
The bottom line is you want to accurately focus, and then have the shutter speed fast enough to prevent camera shake (and you can turn on SR), but with an ISO low enough to limit the sensor noise in the image.