It's a patch - you don't need a special screwgun, you don't need ANY screwgun.
Use screws - nailing drywall is an art form and takes a special hammer. Screwing is easy and takes a Phillips screwdriver. If you use a "not special drywall screwgun" stop before the screws are all the way in and use a screwdriver to finish. The head of the screw should end up just below the surface, but not so far that the paper breaks. Practice a couple on scrap - start with going too far and then you know what not to do - get a couple just right before you start on non-scrap. Backing up after you go too far does not work - the strength is gone when the paper breaks. If your shortest knife run across the surface does not hit the screwhead, it's deep enough.
You don't really need 3 knives/trowels (knife, ala puttyknife, is more common in this context) for a patch, either, though they are nice to have when doing a whole room (same applies to the screwguns, which can be rented when you have a whole room to do.) A big one (12") and a small one (3-4") should be sufficient, but if you want to be over-equipped for this job or equipped for the next one, add an 8". Try a yardsale if you can find one, if cost is an object.
Mud lingo - don't get too concerned, it's a patch, you want standard mud that you can get in a small (4 lb or so) tub. There's rarely a choice in that size. Since you don't say what mud lingo you don't get, I'll go into a few - hot mud is dry powder you mix with water that sets quickly (the number of minutes it's supposed to be workable are usually in the name.) Standard mud comes as a paste in a tub or bucket, and dries rather than sets. Lightweight mud is like standard mud, but lighter (almost foamy.) All "mud" is actually labelled as "joint compound" but pretty much universally referred to as "mud"
The secret to mudding is to get it on good enough and stop, let it dry, and then put on the next coat. Trying to make the first coat "perfect" usually makes it worse than if you stopped 20 minutes earlier. It will shrink when it dries anyway - the multiple coats of mud serve to fill smaller and smaller defects, so there's less fresh mud, so it shrinks less each time.
Just buy a little tub of joint compound, or spackling and cover up the areas. You will have to paint it as well. If it is a hole then get a small piece of drywall slightly bigger than the hole but where you can still get it inside (you may have to cut it bigger). Drill a hole in the middle to fit a piece of string in it. This will help you to hold it to the wall. Use joint compound on the edges like glue. When it dries, fill the void with more joint compound. I've repaired many fist-size holes from when I was a teenager.
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If just the paper on the surface is broken you can fill the gouge and paint over it. If the sheetrock is broken and falling off it is better to cut a section out. I find the closest stud on either side and cut the sheetrock down the center of the studs, cut a new piece to fit and screw it in at the edges of the new piece and the original area cut off then tape & texture to match and repaint.