Oh those pesky holes! I will talk more about what product to use (and not use) in a different blog article so let me talk about technique here.
Using your drywall knife apply the mud in one direction and take off in another direction, less chance of pulling out some of the fill than when you put on and take off in the same direction, try it and you will see what I mean.
Thinner layers are better than thicker layers. Thicker layers take a long time to dry and will usually crack.
If you have a bigger hole you can buy mesh tape or mesh patches. They are a bit sticky, put them in place and mud over them. For these bigger repairs use bigger knives. The less ridges the less sanding and that is always a beautiful thing. These bigger repairs will require multiple coats and each coat should be larger than the last so that you don’t end up with a “wow” in the wall (where when the paint is applied you can see a high spot on the wall where the patch was.
Repairs create an interesting phenomenon. You do 200 repairs, sand and paint and after the paint dries you see another 50 holes that you swear weren’t there before you started painting. Don’t worry about you can fill those fifty before painting but know after you paint the next coat you will see another 10 you swear weren’t there!
Mud always needs two coats of product on it… if you are mudding and then painting two coats you will be good, if you paint a coat then mud some more just go around with a semi dry roller and get a coat of paint on the repairs and then roll on your top coat.
Mud absorbs paint differently than a painted wall so if you don’t get two coats of product on the mud spot it will “flash” which means you will be able to see each mud spot glimmer on the wall.