IMO, it would be a shame to cover redwood or cedar with paint. You can use pine boards, either pressure treated or plain. Pressure treated is generally wet when you get it, and it can warp unexpectedly. However, it's best for in-ground posts. Since you plan to paint I don't see why plain SPF (pine, spruce, fir) wouldn't work for the horizontal boards.

Ideally, i would start by priming all sides, edges and ends. Allow to dry and paint all sides, edges and ends. Then attach these horizontals to the posts and paint over the fasteners. Better to set posts on 6-foot centers or even 4-foot centers, because you won't have the usual 2x4 horizontal backing (3rd photo below) that gives upright fence boards more strength.

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We bought a 4 year old home where the original owner used some sort of pine boards on a fence and painted it. Boards are rotting, the entire fence is peeling and looks terrible. Having to pay $$$ to rip out and replace the whole thing - if you don’t have the budget to do it right at this time, save up and do it right the first time - or you’ll be like us and re-doing it in a few years.

Rough sawn cedar takes solid color stain very well. Painted untreated pine or spf would not last in our climate but might in yours. Douglas fir is a better choice. Any wood that's not treated should get finish on all 6 sides before it's put up.

Thanks for all the comments. I ended up going with tongue and groove redwood which we’re going to clear coat seat. Came out great!