Add a splash of detergent to the warm water and wipe the ceiling down with a sponge to remove any dust and dirt from it. Let the ceiling dry completely, for at least a few hours, before beginning the paint job.
Remove any furniture from the room while you wait for the ceiling to dry. Take any pictures and curtains off of the wall. If there's an overhead light on the ceiling, take it down temporarily. Spread drop cloths over the floor to protect it.
Pour the primer into the roller tray. Dip the 2-inch wide paintbrush into the primer and paint a 2-foot-long strip on the edge of the ceiling in one corner. Apply a strip of primer on the other side of the corner, perpendicular to the first strip of primer. You should have an "L"-like shape painted. Start in either the lower left or upper right corner.
Dip the roller into the paint tray and roll it about to coat it in the primer. The roller should be wet with the primer, but not dripping. Starting in the corner with the strip, paint an "M" shape. Fill in the "M" shape with a "W" shape to avoid streaks.
Paint another 2-foot-long strip of primer on the edge of the ceiling with the paintbrush, then fill in using the primer-coated paint roller. Doing so prevents the edge from drying before you paint next to it. Continue painting 2-foot-long strips of primer along the edge of the ceiling and filling in with "M" and "W" shapes as you work your way across and over.
Let the primer dry overnight. While it dries, clean your roller, brush and tray.
Repeat the process with the ceiling paint the next day. Let the paint dry overnight before you paint the walls or bring furniture back into the room.