My background is retail paint store management 21 years.
A few days before and after obtaining the materials, you'll need to pressure wash the wood, wait a day for it to dry, then apply the primer.
You need to use an oil based primer like Zinsser's Cover Stain, Insl-X Prime Lock. This will block the chemicals used in the pressure treatment from bleeding through the paint film. Acrylic water based primers "might" block these chemicals, oil based "will" block these chemicals.
Then you can use an acrylic water based stain, acrylic water based house paint. I suggest acrylic because oil based finish (final coat) products have a tendency to mildew, they also have a tendency to chalk quicker than acrylics.
Benjamin Moore's Arborcoat acrylic solid stain line (#640) is a good one. Or, if you want even more longevity, go with Benjamin Moore's Aura line. (#629)
If you can rent a sprayer, this will make it easier, otherwise brush and roll. The brush needs to be any Nylon/polyester blend. Roller cover needs to be between 3/8''- 1/2'' Woven or microfiber for both the primer and the finish.
Water based exterior paint, if applied properly under the proper conditions will not wash off in the rain.