Are there any tips anyone can give me? I've never done this kind of thing before and my lens is totally new, so I have not had much experience with it yet.
The main thing here, no matter what lens you're using, is that it will be difficult to use the automatic exposure modes (P, Av, Tv) because the dark sky will cause the metering system to overexpose the moon. If you really want to use auto exposure, you can try spot metering mode to get the camera to evaluate the moon itself rather than the sky. Another option is to use evaluative mode so that the camera looks at the entire scene and then use exposure compensation to decrease the exposure until you get a properly exposed moon. But you'll really be better off in manual (M) mode and dialing in the exposure yourself.
For any shot, the three exposure parameters you can always control are: shutter speed, aperture, and ISO. You want the highest shutter speed you can manage, so reduce motion blur, and the lowest ISO, to reduce noise. Although your lens is sharp at f/2.8, you might want to stop it down a bit for even more sharpness. Here's a moon photo I took recently:
This shot was taken at 1/180s, f/8, ISO 1000 (with a Sigma 150-600mm lens), so start with those parameters and adjust as necessary to make the shot you want. Know that because the moon is so much brighter than the sky, exposing the moon properly will mean that the sky will look much darker in your photos than it does to your eye. If you want a photo of the moon against a deep blue sky, shoot during the day or just after sunset.