Finish carpenter here. 80% of the result is in the prep and unlike walls getting a dead flat finish on mill work and cabinetry is largely about the primer.

1. Any bare wood should be water popped before priming. Ply wood should be fine to do this to whatever the off the shelf sanding level is assuming you used decent stuff. Face frames should be sanded to 150. Wipe its all down with a wet rag which will cause the grain to raise then sand with 180 and clean thoroughly with the last step being a wipe down with mineral spirits or naptha.

2.). If you’re looking for off the shelf solutions then use BIN shellac primer. It dries in 15-20 minutes and sands very smooth. The fumes suck but dissipate somewhat quickly. Don’t use the sprayer for it as those Mandela can’t be used with flammable products. Get corners and edges with rattle cans and the flat surfaces with 1/4 nap roller (no foam). After the first coat fill any imperfections with joint compound and sand thoroughly. It’s ok to sand through it because the first coat is about creating a uniform surface. I do 3 coats until it looks like it’s actually painted. That’s the heft you need. Sand to 220, vacuum and then wipe with wet rag. It should be very smooth.

3.). For paint BM Advance is the best out there and I’ll take anyone to the mat who thinks SW competes. You’ll want to dilute with distilled water by 5% or so to aid in atomization (can’t use floetrol with it). Spray the door backs first and let dry for a 3-4 hours then to the front and edges. They need dry for 16 hours before the 2nd coat (no compromise here). In between coats wet sand with a fine grit sponge - just use a container of clean water and keep dipping a wringing out. It should leave a cloudy surface then wipe clean with a wet rag. Let it dry for 15 minutes then spray the 2nd coat. Be careful with controlling the material and practice/sample to get your flow perfected. Advance is a little on the loose side and can drop if you’re not careful (constantly check for drips and smooth them out with a roller if you find one still wet.

Let it cure for $ days before use and it should be very durable.