Devil's Bridge is a good one. Note that there is a new(ish) preferred trailhead called the
Mescal Trailhead that you'll want to use. The old trailhead was down Vultee Arch Rd/FR152, which is only passable by high clearance vehicles. Some maps (including Google Maps) still show this as the TH.
A couple other good trails nearby are
Doe Mountain and
Boynton Canyon. The
West Fork trail is a good one too, but extremely popular, especially on weekends.
You'll need to get a
Red Rock Pass to park at the forest service trailheads. Daily (and I think weekly) passes are available at the automated pay station at the Boynton TH; also you can get one at the
ranger station visitor center on Highway 179 just south of the Villiage of Oak Creek. West Fork has a private TH with a separate entrance fee.
Devil's Bridge and Doe Mountain trails don't have a lot of shade so they can get pretty warm in the middle of the day. Boynton (except for the first mile or so) and West Fork get less sun so they stay cooler. West Fork follows a creek that you'll have to cross a few times too (typically the water is a foot deep and very gentle- it is possible to boulder hop to stay dry).
Never done cathedral or broken arrow so can't comment on those.