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The previous owner painted over the door frames and stair banister with thick white paint that has shrunk & cracked. What would be the best way to prepare the surface so I can re-paint?
Looks like you have a case of bad adhesion to an underlying varnish finish. I suspect the former owner put on a heavy coat of paint without properly preparing the surface and probably without a primer designed to give adhesion. I would bet it is probably a latex paint also. Does the paint have a "ropey" look with lots of brush marks showing? Does it feel kind of gummy? No matter, to make it look really good, you should remove the finish. I suggest you try to scrape it by baring down on a putty knife and dragging it over the finish. If there is really bad adhesion, often baring down will pop the paint off. That which does not pop off, you might try a heat gun to help break the bond. If that fails, you might try chemical strippers.
It would be best if you got down to bare wood, but if that is too much work, merely sand the underlying varnish well and/or wipe it down with a liquid deglosser. Prime the surface with an enamel undercoater and then finish coat it. I am still an advocate of oil paint for a good hard, brush mark free finish, however, there are now water borne enamels on the market which are beginning to rival oil enamels in appearance and durability.