Help peeling painted cabinets

Ok serious answer. If in fact they are urethane, start with Dawn. If that isn't quite strong enough, step up to Barkeeper's Friend powdered cleaner. It will clean and polish most anything and not scratch the urethane finish.

I have painted my share of cabinets - at least 20 kitchens.

1. Clean everything - I personally use SOS pads because it scratches through oil build up and just knocks things out. Wipe up excess soap with wet towel.
2. Sand - a rough grit. The worse/thicker your old paint the thicker the grit. You will get advice on stripping the cabinets - don't. Not unless the paint is "loose" almost everywhere. And if the paint is "loose" it is usually a moisture problem or really cheap cabinets. This is a quick sanding too. A typical 20 year old mid size kitchen would take maybe 1-2 hours to go over. You are looking to take care of flakes and uneven surfaces. If there is a lot of flaking sometimes taking a wallpaper remover blade to the cabinets is faster then a quick sand.
3. Assess the doors and insides. While sanding you can get a good idea for how the doors could turn out after painting. There is no magic to fix cheap/rough doors. Also if doors have ornamental designs or routed lines these are very hard to paint well - almost impossible to hand paint well. If you need new doors I highly suggest rawdoors.net (no affiliation just a very happy customer with many orders). You get solid hardwood doors with hinges in lots of different wood flavors and you pick the exact size of doors you want - with older kitchens you can often get a more modern look by upping the door sizes. On the insides of the cabinets - you don't have to paint or even repaint these. First if they are the big box cheap laminate looking crap painting them will take forever - they have a clear coat that most primers don't even stick well to. If it was painted before I wouldn't repaint unless the color is awful or paint is deteriorating.
4. Buy your primer and paint. I highly highly suggest using oil based semi-gloss. You can go latex but I have found that it has a very rubbery look and just doesn't look good on kitchen cabinets. Your primer is based on your paint choice.
5. Thin coat of primer on everything you are painting. I normally would use a small foam roller for all painting with cabinets. It keeps the finish smoother and less drips. Spraying is an option too. There is usually flooring or something in the kitchen I don't want to get paint on and I have had no luck in keeping spray dots from hitting things sometimes 10 feet away. Unless I can take the cabinet/doors outside or I am doing a complete gut job, I roll not spray.
6. Fine grit sanding.
7. Second coat of primer.
8. Paint.
9. Second coat of paint.
10. I paint the door fronts again once they are installed - and hit any blemishes.
11. Optional - clear coat the outside of the cabinets and doors.

Color... Almost always a fancy off-white. I tell the paint store what backsplash and what color the walls will be and they pick out a white that looks like a slight variation of... white. I have no design ability and I an usually looking to please the general population so the white variations work.

The last kitchen I did I installed a mid sized island (used cheap lowers from Lowes at $120 total), primer/paint $160, all new doors/drawers - about 30 - (including the island cabinets) $750, and new door knobs $40. So the whole kitchen cabinets looked brand new plus island for a little more than 1k. This is labor intensive but anyone can do it.