When ever I am shooting images of stuff in space it is manual focus always. For things like the moon live view autofocus and regular autofocus (with the AF fine focus adjustment done properly) produce generally about the same results for me. However some times they can be fairly far off. The moon is really bright and even with a longer lens, say a 300mm f/4, it is still pretty small in the scene. Add in that unless you are lucky and get autofocus to focus (either one) on the edge of the moon it is fairly low contrast. Add it that it is pretty bright (start point of exposure is the loony 11 rule or ISO 100, f/11 1/100s) and it is fairly fast moving and getting a good autofocus is mostly a crap shoot.
So what I would do if I were you is instead is find a bright star or, if using a wider lens, planet (Jupiter is up now and should be fairly close to the moon tonight) and use magnified live with the camera pointed at that bright point light source with focus peaking turned off. Now manually adjust focus until you have minimized the star or planet (only use a planet if using a lens wider than 100mm) and you will generally be very close to a perfect infinity focus. After that recompose on the moon and shoot away. If you are serious about astro shooting I suggest getting some
bahtinov masks that will work for you lenses as you can easily achieve a real perfect infinity focus with them. For regular camera lenses
I have these (in all 3 sizes), and yes they do work. They are inexpensive but we know how to take care of things right? The fine grill/mess on them is what allows them to produce good results with short lenses and I have used them with a lens as short as 35mm and still gotten visible spikes in magnified live view.
From there when shooting the moon start with exposure similar to loony 11 and you will find that you my need to add 1 to 2 stops of exposure but it is a good starting point. Stick things on a big tripod and use the 2s mirror up delay shooting and a remote release. Beyond that shoot lots of shots of the moon as the atmosphere is rather turbulent so by shooting lots of shots you can pick the best or combine the best parts of each to get a really great shot.