Sherwin-Williams has a range of exterior acrylic latex paints that should work for you. Some of the higher-end paints are even self-priming, so that's another option for you. What is the door made of?
Carolyn, the old loose varnish needs to come off first with some serious sanding. New paint, even good primer, will not soak through the old varnish and stick it down. In fact, it will eventually pull it off and you will have more peeling. Perhaps you can use an electric sander on the wide flat parts.
In the perfect paint world, oil base primer works best on raw wood and latex primer works best over other paint. If you don't want to use oil, latex will suffice.
When you go into SW, ask them which one of their many primers is good for your exact circumstance. ie, if you are only going to use it outside, then don't get an interior/exterior primer. Also, some of their paints do not come in quarts, so if you only need a quart, your selection will be more limited.
You have us stumped with this one...I can't imagine varnishing a fiberglass door but I'm sure people have tried it. If the varnish is peeling off...getting that removed is the first step. Sounds like a wooden door in which case you need to sand it back down, prime and repaint...if a painted look is what you are after. If not, you can sand off the varnish, restain and then apply marine spar if it is a wooden door. Let us know if you still need help!