You don't say if you are talking interior or exterior, but I would guess interior from the staining description - this is from dirty water seeping through the cracks at the panels tothe interior.
You can get more detailed instructions on the web, but here is the short story - I am assuming wood door:
1) for exterior, clean with grain with about 120 grit sandpaper to remove any roughness or splinters, scrub with TSP and rinse, let dry THROROUGHLY for at least 2 non-rainy days, use PAINTABLE siliconized latex caulk (20+ year type like GE) to run a bead of caulk along all the joints between the door frame and the inset panels, after 2+ days drying if badly stained apply a coat of Kilz Exterior paint otherwise a good quality exterior primer, let dry at least one full dry weather day, then 2 coats per instructions of good exterior trim paint. I set the door at a height (part-open) so it is easy to paint each panel and get the edges of the panel a bit, then let it dry (or help with hot air gun on low) to tack free, then spray with silicone lubricant spray to it does not stick when the door is closed. I have had unbelievable performance from Weatherbeater Latex Satin House and Trim Paint (made by Sherwin Williams, I believe) - 20+ years and still looking good though more like a flat now. Then wash with clothes washing detergent and a sponge quickly rinsed off with a hose every year or two. Don't forget to do trim same way, same time to match. Remember you have to keep door away from the sealing face while drying - about 4-6 hours after painting, either by leaving open (like on a weekend when you are home), or if necessary close it but immediately slip nails or small shims in between the door panel face and the stop strip to prevent sticking. REMEMBER to remove shims before opening door - unplug opener to be sure they are out before opening. You can use a brush with easy-flowing latex, or a mini-roller (3 or 4 inch) which does the panel nice - brush to get all the recesses and edges, then do the panels with roller last.
2) for interior same except no caulk - you only want the exterior caulked - leave the interior uncaulked so any moisture can evaporate, sandpapering probably not needed, do the sandpapering and washing at the same times inside and out, and silicone spray is only needed on the outside joint face of the panels. I recommend still using exterior grade paint on the inside and I like a semi-gloss as it reflects light and repels dust better. Also, for full stain covering it may take 2 coats of Kilz if heavily stained. Be sure to tape hinges and roller axles so they don't get paint clogged, and you need to use a thin brush or rag, or better yet compressed air if you have it, to blow the dust and spiderwebs off and from behind the bracing so it does not get mixed in with your paint. I do NOT recommend removing the stiffener bars - too easy to get a warp in the door so it will not fit like it did before. I use brush for the entire inside because with the bracing and hinges there is not much roller-free space anyway. Just avoid the temptation to go back over what you have brushed on one and you should not have brushmark problems. Work across the panels from top one on down, not up and down.