Just put up a new fence in mid July that we checked for moisture, the gadget we used showed zero so do we still need to wait a full year to stain it or can we do it right away? Also if we do need to wait will it still stain up nicely after it's been in the rain all winter? Won't it end up with moisture in the wood?
Depends on the surface porosity, not moisture. See here about new wood: https://www.deckstainhelp.com/staining-a-new-deck/
It applies to a fence as well.