For that small of a job, if i didn't think I'd use it again, I'd just use a cordless drill, or even a variable speed corded 3/8" drill. I did a couple of houses with a 3/8" drill before I could afford to buy a screw gun, and cordless drills were pretty uncommon. It's hard on the drill, but they can be had for $10 or $15 at yard sales pretty readily. Just watch the work as you go, and don't oversink the screws or leave them proud. Just below the adjacent surface, so you can mud over them readily.
Buy 3 drywall knives, and a surfplane for the job. I'd buy a corner knife also, for inside corners, although you can get by without one at the cost of about an hour of labor per corner. Use the surfplane to bevel edges or clean up the edge when you break the sheets for a joint. Use a 3 inch knife to embed the tape and fill the joints. What you want to do with the embedding coat is just get the tape thoroughly adhered and covered. Try to just go over the tape once with one long stroke; don't go back and try to smooth it more. The more you try to smooth it, the more irregularities you'll put into the mud.
After the mud is dry, use the 3 inch knife scraping at a very low angle to knock off the high spots and "boogers" that stick out from the surface. Then, go over the seams with a good generous coat of mud and the 5 or 6 inch knife. This knife will ride on the surface of the sheetrock adjacent to the seam, and alllow you to fill the seam essentially flush with the adjacent sheetrock. Again, one long stroke, don't go over it multiple times. It will just get more irregular the more times you go over it. What you are doing is sequentially getting it smoother and smoother. Use the knife to coat all screw holes now. (you can do it with the 3 inch knife, but it's less flexible, and doesn't work as well).
Again, let it dry and scrape off the irregularities. It is now time to use a 10 or 12 inch knife to smooth out the joints and fill the irregularities. If you need to, do a coat to fill irregularities and places that didn't come out quite right. Let it dry if you do, before going to final coat.
For the final coat, use thinner finish mud and the 10 or 12 inch knife. Use long, even strokes to put on the finish with as little chatter as possible. Use a generous coat of mud to float out the irregularities and taper to the edges of the sheetrock paper outside the previous joint coverage. If you hold the drywall knife on a flat surface, you will find that it is cupped one way. Orient the cup towards the sheet, so the final surface is bulged slightly. That will account for shrinkage, and tapers the edges to look best.
After this is all done, scrape off irregularities if needed, or sand if you have to. Touch up gouges and irregularities with more finish plaster.
I like to put on a light splatter of orange peel; it seems to make the surface more resistant to scratching and gouging, and covers up a little bit of irregularities in the taping of the joints if you're not highly skilled. If you have the ability to borrow or rent a splatter gun, it's easy to do.
Good Luck!
