Painting a tray ceiling poses a few challenges that painting a flat ceiling doesn't. It's still a doable DIY project with a bit of work and patience, though.
Step 1 - Prepare the Area
Remove any furniture you can from the room. Put drop cloths over everything else. Then, put drop cloths on the floor. Make sure they overlap and are taped one over the other. Remove any curtains or draperies.
Use your vacuum cleaner with the dusting wand extension to remove dust and cobwebs from the ceiling. If the ceiling has stains, cover those areas with an oil-based primer to keep them from bleeding through into the new paint. Finally, remember: you can use an oil-based primer with a latex top coat but not the other way around because the paint won't stick.
Step 2 - Paint the Horizontal Section
Put a paint roller on an extension handle. Pour ceiling paint into the paint tray. Dip the roller into the well end of the tray and use the other end of the tray to work the paint into the nap. The roller should be saturated but not dripping.
Start at the far end of the ceiling and work your way to the other end. Some people like to paint in a "V"- or "X"-shaped pattern to avoid roller prints and get good coverage; it's up to you. Allow the paint to dry. If you are using different colors for the horizontal and vertical sections, make sure the horizontal coat goes all the way to the edges and perhaps a bit onto the vertical surface.
Step 3 - Paint the Vertical Section
If you didn't do so for the horizontal section, you'll definitely have to use a ladder to paint the vertical sections. Put some paint in a small bucket and cut in the edges with a brush. After that, start painting at one side of the room and work your way around it. Lay on a good smooth coat of paint so you won't need to re-coat this area. If your ceiling has more than one tray or other decorations, use the brush method for those as well.
It's easiest to paint the entire ceiling the same color. Otherwise, be careful to create a straight line between the flat ceiling and the vertical edges just as you would between a regular ceiling and walls.