Wipe then down with tact cloth before starting to paint. If I were you I would go to a real paint store & take your list of questions to the experts. They can show you what supplies you will need. Myself I would use latex enamel. As long as you give them enough time to cure they will be very durable. I used it in my pantry 24 years ago and it has held up well. That includes the shelves.

I would use a cabinet paint on them, which will be a lot more durable, after it has properly cured. A good cleaning, and then a good sanding, and then a good wipe down with tack cloth to remove all the dust, and then a good coat of a bonding sealing primer. Once the primer is dry, sand very lightly by hand with sandpaper. Do not remove primer, just sand it very lightly, then put at least 2 coats of cabinet paint on, and follow the directions for the brand you choose. Some say 24 hours between coats.

Every thing you asked about and that has been discussed is meaningless unless you identify the existing finish. Failure to take that into consideration when windows were painted is the reason they are peeling now. Those look like adjustable shelves,take one to an expert to have old finish identified. A door could be removed if shelves are fixed. Once existing finish is known,correct preparation and primer can be chosen. Some old finishes are simple to paint over(if you know which it is) while others require heroic effort to properly refinish. Knowing this is one of many issues real painters know about,hence they cost more than minimum wage.

All the major paint manufacturers have paints suitable for cabinetry re-coating. Those I know of don't require a dedicated primer, but merely two coats of the paint.

The most important thing is that they be clean, especially of oils from human hands. A scuff sanding with 220 grit sandpaper is advisable. If you use TSP for cleaning, rinsing is very important. If the old finish readily dusts up when sanded, you will not any exotic type primer, to which the above post refers. If adhesion adhesion is of great concern, nothing beats BIN white pigmented shellac as a primer.

Spraying will always give the optimum results, but good results can also be had using a trim roller and brush with these dedicated cabinetry paints.

No TSP. TSP needs to be flood rinsed, plus it's overkill for a project like that. Dirtex is a good prior to painting cleaner. 3M sanding blocks medium grit are good for sanding and if you use sandpaper, 100 to 150 grit is good. You just want to sand the sheen to dull. I think Benjamin Moore Advance is the most durable paint you can get. You will want to wait a couple of weeks at least before putting heavy objects on the shelves since any latex product takes time to cure. This is why we get sticking. Objects sink into the soft paint film. Over time, it gets harder.

The problem with TSP is that it turns oils and grease into a form of soap. It is not good to paint over soap either! It doesn't have to be flooded, but just rinsed. If there is an advantage to SP, it is it too dulls hard varnishes and enamels.

For general cleaning, i agree with paintguy 22, Dirtex is a fine product that doesn't need all the rinsing that TSP does.

I do feel that 100 grit sandpaper is too rough, possibly leaving scratches which might show through the new paint film.. 180 or 220 will dull, but not scratch. I prefer cut sheet sandpaper to sanding blocks. Most flat cabinet surfaces are not as flat as you think.,scalping of high spots and missing of low spots can result. Also, blocks don't get into machined detail like ogee routed edges.

Fabulous results!

Very nice job on them. You can be very proud. I really love the turquoise color. What color is it?