How do I prep and paint treated lumber for an outdoor trellis?

Don't paint treated lumber until its dried out. It may take as along as 90 days for that to happen. Try to pick out dryer treated lumber to stat with. Many times the wetter is straighter to use and that's what I usually end up using.
Build the trellis so the lumber doesn't warp sitting around to dry.

Once its dryer out use a good ext primer sealer and good paint. When its cost effective I will get my primer sealer tinted close to the finish coat. ( less coats better color achieved)

Signed 21 Boat

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21Boat is correct in that you have to wait until the wood is dry before painting or sealing. Wet wood is usually straighter than treated wood that has sit inside in a stack and dried (indoor lumber yard/store). The drying time can vary with the amount of direct sun, heat and air flow. 90 days may be good for what you are building. More for a deck may be needed. Painting wet wood will seal in the moisture - this is a very bad thing. This moisture will still work its way out THROUGH the coating. This will ruin the entire paint job. Before painting, inspect all surfaces for defects and rough areas. These should be fixed or sanded prior to painting. Use a high quality coating since this will be exposed to everything nature has in your area. A high quality coating should last 10-15 years if done right.

In some parts of the country, you can purchased "Kiln Dried After Treatment" Pressure treated wood. In that case, you could paint right away. There's also varying levels of treatment, with some approved for ground contact, some for below ground or wet service and some treated only enough to withstand the intermittent elements, such as railing and decking boards.Be certain to purchase the proper PT for each aspect of the construction.