Lay out the drop cloth. Put the Billy bookcase on the drop cloth.
Lay out the two slats in parallel fashion. Remove the removable shelves. Lay the shelves across the slats. Placing them on the slats keeps them off the floor.
Sand the bookcase and the shelves with the sanding block. You don't need heavy pressure. Firm strokes, without pressing hard into the bookcase, will be sufficient to prep the surface.
Moisten the rag. Wipe down the bookcase and shelves. Let them dry a few minutes.
Brush on the semigloss paint starting at the top of the bookcase. You won't be able to cover the Billy bookcase adequately in one coat. Count on two or more light coats. For this reason, paint thin, smooth coats. Follow the lengthwise directions of shelving with your brushstrokes.
Paint the shelves you have removed on the top and sides. Allow the shelves to dry. Flip them and paint the other sides. Watch out for drips on the edges.
Repeat the painting process, applying a second coat. You may need three full coats to cover the bookcase satisfactorily.