If you like the wood, I'd just leave it natural. Mix up a batch of linseed oil and mineral spirits (50-50 or even 75-25) and paint it on liberally. I like to use those cheap disposable paint brushes because it gets pretty messy. The mineral spirits will help the oil absorb into the wood. Wait 15 minutes. Using a rag, wipe off any residual oil that has not absorbed. If it ALL absorbed (and it very well might since this is pallet wood and could be quite dry), then I'd put on a second coat. Repeat the wipe-down after 15 minutes. If the wood is rough and hard to wipe down, just do the best you can. If you don't wipe down the residual, it's not the end of the world, it just gets tacky to the touch when it's dry. It will darken your wood slightly, but also bring out the grain. You're done.
Both linseed and mineral spirits are easily found in the paint section of your hardware store. Linseed oil will protect the wood, moisturize it, and also provide a waterproof barrier from inside the wood rather than just coating the surface like Thompson's would. I find that Thompsons will peel up pretty quickly. I oil all my teak and cedar patio furniture every spring and after 10+ years, it still looks like new.
I personally painted the interior of my coop to make it brighter and easy to clean. Absolutely no regrets. Prime first, maybe even 2 coats, and use an exterior paint (latex or oil, doesn't matter) to better manage temperature fluxuations. Do the floor while you're at it. Late last summer, I used waterproof stain on the exterior and was sad to see that after only ONE winter it was dry and peel-y and had to be redone. So I used the linseed this spring and am much happier.
ETA: After re-reading your original post, it sounds like you are looking for advice on protecting the wood and then painting? Don't seal the wood with Thompsons first. It will hamper the adhesion of the primer. Just use a good quality primer - I prefer to use oil-based primer when working with wood and I usually use Kilz - and go with multiple thinner vs. singular thicker coats. You can use latex or oil paint over that. Again, thinner multiple coats. Make sure your job is completely dry between coats. Shouldn't be hard in Arizona. Good luck!
