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Most stained doors were done with oil based coatings. That means you need an oil based primer first then you can use latex paint. Latex primer will not adhere to oil based coatings.
Dennis D. Gehman
President
Gehman Design Remodeling
PA297
355 Main Street
Harleysville, PA 19438-2417
Office 215-513-0300
Cell 267-718-4697
Fax 215-513-1280
Karen, I'm not a painting specialist, but from my experience you would be better off with a good coat of oil based primer. First, sand thoroughly, including the details, then wipe clean with a tack cloth or lint-free rag dampened with solvent.
Please do yourself a favor and don't beleive "paint and primer in one". All that refers to is painted surfaces that have minor stains or a large color change. There is a reason that every paint store carries numerous primers. That being said, I echo the thoughts of the previous participants and say to use an oil primer that is specifically a stain blocker. These smell very bad but the smell will dissipate quickly and can usually be recoated in an hour.