Linda...sand well..then clean
I would start by sanding the cabinet. Next I would clean it. Once those 2 steps are done, I'd paint it with chalk paint. You'll have the option to seal or not with chalk paint, but I recommend sealing with a top coat.
Linda: do NOT paint without sanding or stripping. You will be so disappointed -- and your hard work of painting will be wasted. (Also take off hardware for painting, for a professional look.)
Sand them first
So, you probably don't want to hear this, but all loose and peeling paint/varnish must be removed. Either by sanding or stripping with a chemical stripper. Then prime and paint. The prep is always the hardest.