I recommend you use TSP cleaner first and then rinse them well with clear water. A light sanding and primer should make the paint adhere well. You will probably find that a semi-gloss will be the easiest to clean-up in the kitchen.
Ditto on giving them a very thorough cleaning. You will not regret it.
I recommend taking off the drawer fronts and cupboard doors (number everything so you know where it goes) and removing the hardware. If I had my druthers, I'd strip every piece of painted over hardware in my house. Keeps the hinges especially looking sharp and working properly
Yes, be sure to sand, clean and use a high quality primer before painting. When you get ready to paint buy high quality paint. Watch out for "reds" as they do not seem to cure (remain sticky) as other colors. Also let the paint dry very well between coats. Don't rush the process. Good luck!
You might also check with Franks or Jacks Auto Body shops about what it would cost to have them paint them. They can put on a lot harder smoother finish then a home job. They would also give you tips on how to prepare them before they spray on the final finish.
They do make a trim and cupboard paint, it's more "heavy duty" than regular paint... you have to wait quite a while for it to dry, and at least 2 coats is good...mine could have used another, in places where you grab a lot or in front of the sink where you stand all the time is wearing off...
I think spraying the paint is your best bet for a smooth finish. If you're going to brush or roll it, use a paint that is made for trim and cupboards or use a paint additive like Floetrol to help smooth the finish.
Lots of great advice has already been posted. If the wood is in good shape you could get a new look using gel stains. The current trend is very dark almost black stain. If you have unpainted stained wood trim throughout house stain cabinets one shade darker. My personal preference on wood stain is very rich teak. If you are committed to painting, I would paint uppers same color or one shade darker than wall color and lowers dark. Remove fronts drawers and use paint sprayer on them. If you do not have one use a very skinny foam roller. In order for good hard cure on paint the air and surface temperature need be plenty warm and not humid. A circulating fan helps process. Always use best quality paint you can buy. Take lots time to prep surfaces and do not rush drying times between coats. Good luck.