Well, Bob, I've remodeled 7 houses and hung a lot of drywall.
The first thing I'd recommend is to use screws and not nails. And use a drywall screw driver you can put in your drill or electric driver. Get the kind that has a clutch that disengages when the screw has gone in the proper amount.
Something like this:
Dimpler 16627 by Vermont American | DoItYourself.com
For a small job like what you are doing, a bucket of mud from the hardware store should be sufficient. I like old fashioned paper tape, others like the self-sticking kind (I don't because I think it can peel after a few years).
Make sure the mud is clean and uncaked. It's cheap so feel free to toss out a batch that has dried grit in it. It will save hassles of trying to smooth things.
Use a smaller trowel to spread the mud for tape application. Then use progressively larger ones to smooth it. For this size job you'll probably get the hang of it about the time you're done.
It's easy to cut drywall with a utility knife and a straightedge. Cut one side, bend the two sides away from the cut, and then cut the other side though the bend. I'm sure the video goes into all that.
I like to sand the joints with a hand sander that has a vacuum attachment. Hardware stores have them specifically for drywall. The dust is horrible otherwise.
I'm sure there are all sorts of other advice, but just do it. It's fairly easy to fix if you mess something up.