I have worked with plywood before and I think I know exactly the problem you are having. Plywood it a lot more uneven and porous than MDF or particle board, so to achieve a near-perfect, laminate-like finish, here's what I do:
1. Sand it smooth to the touch, I like to use a coarser paper, 150grit works well. Use an electric orbital sander.
2. Get a big tub of carpenter's wood filler, it has to be water-based. In a separate plastic container, mix some of the filler with water until the consistency is like pancake batter. Skim each side of the cabinet with the thinned-out filler (you will want to have the side you are coating facing up so gravity can do it's job to level it out).
3. Go play some games on the PC and makeshift control panel you have on the workbench while you are waiting for the filler to dry, then flip the cabinet around/over and do the other sides.
4. Sand the filler until it is smooth to the touch, use the electric sander with 220 grit paper.
3. Roll on some latex primer with a cheap dollar store roller then throw the roller/paint tray away
4. Go tweak the trackball settings and .ini files on the PC and makeshift control panel you have on the workbench while you are waiting for the primer to dry
5. Lightly hand-sand the primer smooth to the touch with 220 grit paper
6. Spray on a guide coat of rattlecan primer. The cheap dollar-a-can automotive gray works well. For a full-size upright cabinet you will need about 6 cans.
7. Go back to playing games/drinking beer until it is dry.
8. Look for any imperfections in the wood (there shouldn't be too many by now) and fill them with the putty knife and wood filler (no need to thin it). Let dry, sand smooth with 220 paper, and touch up with primer.
9. Apply the first finish coat of paint - if you plan to roll it on, use a high-quality, satin-finish latex paint and a good quality foam roller (not from the dollar store). If you want to rattlecan it, use Krylon satin or semi-flat paint FOR METAL, get it at the auto parts store. The Wal-Mart Krylon is crap.
10. While the paint is drying, run up to the liquor store because you are most definately out of beer by now.
11. Lightly hand-sand the cabinet with 1200 grit paper and apply the final coat of paint.
It sounds like a lot of work...it will eat up an entire day, but that is the process I use to get a perfect finish on plywood.