in a kitchen the surface is likely to have a film of greasy dirt.
I wouldn't use a water method such as sugar soap. White spirit will clean it off, you will see the dirt coming off on your rag. When you clean it again and get no more dirt, it's clean. You might find it then doesn't need repainting...
it will probably be some kind of hard sprayed laquer. Wipe it lightly with fine sandpaper to remove the shine and give it a key. Wrap the sandpaper round a cork block. You can sand harder on the solid frame than on the veneered centre. Then clean it again with white spirit to get the dust off. You may find an oil-based eggshell sticks OK, give it a try on a small piece. If not you will have to use an oil-based undercoat.
There are water-based primer-undercoats but they do not stick as well, especially to a laquered surface, and I find them softer.
A gloss finish will be easier to keep clean than eggshell.
Apply thin coats and wipe very lightly with extra-fine sandpaper between coats. And yes, wipe the dust off with a clean rag damped with white spirit. Wear plastic or nitrile gloves, it will eat through rubber.
I would use a smallish soft pure-bristle brush. There will be too many corners and edges for a roller. Do the edges first. If you are doing the insides of the doors, do them before the fronts. Take the doors off and stand them on tins so they don't touch the newspaper you have put on your worktop or workbench.