I have built in cabinets in my bedroom and recently we painted them. I have used at least two coats and yet I still get these spots after it dries like I missed somewhere when I know I haven’t. To add to my problems the paint seems to chip or peel very easily. What can I do?
Fixing your paint chipping problems permanently isn't possible without stripping off all the paint and starting over. At this point you will have to maintain your current finish.
The "bleed through" spots and chipped areas will need sanding down to the original finish or a little beyond. Paint chips can be "feathered" into the surrounding areas while sanding. All raw unpainted areas need oil based primer to promote better adhesion.
Kilz Original oil base primer works good. Apply 2 coats with an hour between each coat of primer. The final coat of Kilz primer can be lightly sanded after 4hours for a smoother appearance.
You can touch-up these areas or repaint from wood joint too joint. Doing just the touch-up could cause some "flashing" problems with the sheen or color, but it's worth trying.
Your problems all stem from the lack of adhesion of the first coat. You didn't mention if the cabinets were previously painted or stained and lacquered, plus the amount or type of preparation you did before painting. Usually it's the preparation or lack of that causes most problems.
Paint adhesion problems can be caused by furniture polish, not priming an oil base finish before applying latex or acrylic paint over it and body oils around door and drawer pulls. Cleaning, hand sanding and the use of a deglossing compound, like Paso, is the only way to get rid of the surface contamination.