Door stains: fixable?

I saw that I need to prime the wood before painting, what wasn't mentioned was what primer I should be using. Finish product is going to be a lacquer paint.

I’m looking to repaint my stained kitchen cabinets. I need advice on how to do this the most economically. I’m looking to use DecoArt for the solid wood cabinet doors and countertop trim because the paint is thick and covers well. I’m uncertain what kind of paint to use for the fake wood laminate. DecoArt paints are pricey, and if I can, I would like to use a different brand of paint for the larger areas. By doing so, it would cut the cost considerably. [img]https://cimg5.ibsrv.net/gimg/www.doityourselft.com-vbulletin/360x480/img_20220306_083045246_0dfd98214dfcdb8f3d64303f35455dcff26fa6e6.jpg[/img] [img]https://cimg8.ibsrv.net/gimg/www.doityourselft.com-vbulletin/360x480/img_20220306_083108990_f2a87e2db9b69c5a2af7728e59af1f4ec11b1aee.jpg[/img] There are also inserts in the doors that are a weaved pattern. I was thinking of turning them around to the flat wood backing and paint the flat side. What would be the easiest way to remove the inserts, which are wood glued in place, without damaging them? Thanks.