Of course you can paint over stained wood. It will probably have a poly or varnish finish on it though so you will need to rough that up with fine grit sandpaper.
And you must be diligent about cleaning kitchen cabinets before trying to paint them. TSP is the paint industry standard. I actually like Purple Power and a clean rinse. You can get it at auto parts stores. I start with low odor mineral spirits first.
I like priming kitchen cabinets with a paint store, not a box store, alkyd primer but you can use a good superbonding waterbased primer too if you plan to finish with latex semi gloss or something which would be my choice these days. You will want two nice finish coats.
Use good tools and add some Floetrol to the paint.
You want a slow drying oil primer, one that takes 24 hrs to dry.the fast drying oil primers like kilz, dry so fast they become brittle and dont hold up in the long run. The slower the dry time the more time it has to bond to the surface for a stronger base coat that will resist chipping better down the road.
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