People are making this out to be much more complicated than it needs to be. Regular drywall (especially for such a tiny patch) is fine-- your box store likely sells 2x2 sheets of the standard stuff so you don't have to wrestle with an entire 4x8 sheet of green board on your way home. A small tub of regular, premixed drywall mud is fine too.

Regular paint and primer is fine unless you have moisture issues (e.g., no exhaust fan).

Get a small (3-4") drywall knife and a medium (6"). The 3" will be useful for your initial coat of mud (and is generally useful to have around the house for patching nail holes, etc.), and the 6" for your second and third coats. You won't need a mud tray for such a small project-- just work directly from the bucket of mud and use your 6" blade to scrape your 3" blade clean, and vice versa. Be sure to rinse them clean with water and dry them when you're done.

Buy a sanding sponge-- it's hard to oversand with them. I like the 3M seafoam green colored one (the various grits are color-coded).

Good luck!