Use your vibrating sander to take off the top layer of existing paint or finish from the trim. You don't need to strip it down to the bare wood, but it should be completely smooth when you run you hand along the length of the trim.
Wipe down the trim with a rag soaked in mineral spirits to take up all the dust.
Lay painter's tape along the edges of the trim and on the surrounding walls and floors, so only the trim itself is exposed. Lay newspapers on the floor under it.
Open and stir your paint. Dip your brush into the paint about an inch. Apply it directly to the trim, running the tip of the brush along the top span. Spread the paint completely, going back repeatedly to brush out any drips.
Dip the brush again and continue applying paint in the same manner, going with the direction of the trim and working from the top down. Cover all of it, brushing out drips as needed.
Let the paint dry for at least four hours. Apply a second coat in the same manner, if necessary. Let it dry overnight.
Run a thin, smooth bead of caulk along the edges of the trim, everywhere it meets a wall, floor or other surrounding material.