Remove the doors from the frames, remove all hardware, sand them to remove the gloss and lightly fill where the veneer has come away from the door. Lightly sand the fill. Use a very sticky primer such as BM. Then paint with a high quality latex. Use a brush if you like the brush marks or a foam roller or brush if you like a smoother look. If you like the look of chalk paint you could use Annie Sloan paint. Huge job but well worth the work. Good luck.

If the wood grain covering is paper thin, any sanding will have to be delicately done. Try taking a drawer front down to the best local paint store and talk to their experienced staff about preparation, primer and paint.

You can also practice on the back of a drawer front after removing it from the drawer.