You have different ways to remodel this kitchen by redoing a cabinet and not everything would be a simple matter of coating paint. Let me explain this a bit further:
If you find your cabinets greasy and dirty,(as is your case) using good bleach would be the initial step for giving these worn cabinets a brand new appeal. A solution of linseed oil, bleach and boiling water will clean your cabinets thoroughly make their appearance very bright. If you see that bleaching is just not sufficient, you can strip off the old paint of finish from the wood cabinet and coat it with new finish. If you wish to retain the color of grain and wood, you can always have the wood stained and apply a single or two coats of polyurethane for protecting the finish.
If you wish to have a complete new look, choose the paint color which would go well with your home decor and pick a brush. Using special techniques for painting like rag rolling or sponging, you can add texture and dimension too for your cabinets.