Forget the Kilz, go to a real paint store like Ben Moore or Sherwin Williams or any other and get what they recommend. I would go with Fresh Start and one coat of primer is all you ever need( with quality primers) two is just a waste or time and $$

Krylon has all you need to keep it simple. ColorMaster Paint and Clear Polyurethane Coating. I'd apply three thin coats of paint followed by three thin coats of poly.

Sand, sand, sand, prime, sand, prime again, lightly sand again and then you should be good to go. Good luck, you will love going white. ~boxerpups. a few before and after going to white

What is the interior of the cabinet now? My stained cabinets have a natural maple interior and I would be inclined to leave that -- except for the blue/grey splash tile. White seems stark. Perhaps look at a color in the tile so you have contrast, but no so stark? You can always add LED lights that are battery operated if you find you need light but can't or don't want to wire them in.