Well...there is a LOT to cover with this.
1) What type of paint are you using? Some paints don't adhere well or can ruin the foamboard. Spray paint propellant can sometimes eat away at the core of the foam, causing it to fall apart and ruin a plane; brushing on acrylics too heavily can also have an adverse effect, causing the foamboard to warp.
2) Are you using the water resistant foamboard, or some plain white foamboard you bought at the local Dollar Tree?
If you are using the brown, water-resistant foamboard, you'll want to scuff it lightly with something like a piece of steel wool, fine grain sandpaper, or even a scrubbing sponge, so that the paint will adhere.
If you're using plain white foamboard, I recommend using a very thin layer of polyurethane to seal the plane before painting it - this will help prevent the foamboard from being eaten away. Lots of people have suggested Minwax polyurethane, which works great; I personally have been having a hard time finding the non-water based version here in San Diego, at least in something smaller than a gallon size bucket. My local Home Depot has Varathane Ultimate Polyurethane, in a half pint container, which was more than enough to cover my Master Series P-38, and I still have half the container which I plan to use on my P-51 Master Series before painting. It comes in a can like this, for about $7-8, I think, maybe less?
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And yes, you are adding weight when you apply paint and Minwax on a plane, truthfully, it's adding only a few grams of weight, and it's never been an issue for my planes. If it's increasing a plane by 200-300g, you're applying the polyurethane and paint WAY too thick. You want to apply it SPARINGLY, and VERY lightly.