You can use spackle alone for what you are doing.
What you should do is buy a roll of mess drywall tape (just look anywhere online), get the smallest and cheapest roll because for fixing holes like you are doing a small roll will last forever. Its a mesh tape with holes in it, so cut the tape about an inch larger and over lap slightly multiple pieces if necessary to cover the hole. Its not really sticky but doesn't need to be. Then buy a small tub of joint compound, take a putty knife and cover it good, the tape should not show through. Let dry for a day and you are good to go. You can then sand it a bit and paint it to make it match good.