If you do not have adults in the coop, your chicks are old enough to go in the adult coop in your climate. Just put them out there and leave them locked in the coop section. It should be bigger than the brooder so they will love it plus they will consider it as home after spending a week locked in there. As long as the run fencing is tight enough that they cannot squeeze through the openings you can let them out after a week in their coop.
Once they are fully feathered out (and yours should be at that age) rain really doesn’t bother them. Mine love to walk in the rain, probably looking for worms that the rain forces out of the ground. Provide a place they can get out of the rain if they want to and they should be fine. If the wind is blowing they generally don’t like that but just a rain, no problem.
If you do have adult chickens, you have an integration to do. From your post I don’t think you do but if you already have adults out there get back with us and we can discuss that. As long as you have a lot of room it should not be a big problem but there are some tricks you can use to help that process along. If space is tight, then you have to be more careful.