Primer is only for bare surfaces, once something has been painted you
can use regular paint in the future.

If the ceiling is not too dirty you don't need to do much cleaning.
Just check for cobwebs that sometimes go overlooked.

If the ceiling is in a bathroom or kitchen I prefer to use a
semi-gloss which makes it easier to clean. (In a bathroom a paint
formulated to retard mildew growth is best.) Otherwise, a flat white
is normally the choice.