This task really depends on your budget and long range plans.
If the paint is chiping like this it probably is layers on top of layers of oil based paint. And I would speculate that the varnished wood was not etched or lightly sanded and even primed for the next coat of paint. The paint chipping off so nicely tells me that the bare wood might be too slick or smooth for the paint to properly adhere to.
Eventually you will probably have to strip the paint down to the wood as more and more layers of paint just make the situation worse, but as a band aid approach, you can chip all you can then feather out sand, then spackle to fill the void then prime and then paint again. Good quality paint will also make all the difference, I mean paint from a paint dealer.