Use Setting Compound for Big Holes
It's fine to fill screw holes and other small wall dings with patching compound, but for dime-size and larger repairs, and for holes that are deep, it's best to use a joint compound that sets up by a chemical reaction. These are available in powder form with setting times ranging from five to 90 minutes.
The reaction starts when you mix in the water, and the compound hardens in the specified time. The five-minute version is nice because you can buy the powder in a convenient 5-lb. box, and the compound hardens quickly, so you can apply another coat right away. Remember, setting-type compounds are harder to sand than regular patching materials, so make sure to strike them off flush to the surface when you fill the hole. You'll find setting-type compounds wherever drywall taping supplies are sold.
Make a Dent for the Patching Compound
When you remove a nail, drywall anchor or picture hanger, there is usually a little ridge of old paint or drywall sticking out that's hard to cover with patching material. The solution is to make a dent over the hole, and then fill the dent. Most good-quality putty knives have a rounded hard plastic or brass end on the handle that works perfectly for making the dent. The rounded end of a screwdriver handle or the handle of a utility knife will also work. Press the handle against the hole and twist it slightly while applying pressure to dent the surface, or if you have good aim, use your denting tool like a hammer.