@KiwiChic,
i'm not a pro, but i done some painting. (and made paint long ago, but...)
i would do as you say, but prime any bare wood with a good solo primer first, at the least.
actually, i'm kinda a perfectionist and would prime it and then paint it, but you don't wanna hear that, so.
anyways.
good luck, and don't get it on your cat, they are really tough to get clean.