Nearby
Kodachrome Basin State Park is an interesting stop, particularly late in the day if you like photography.
A nice drive (without the trailer & as long as it isn't wet) is
Cottonwood Canyon Road. A short drive to see Grosvenor Arch, or, with good conditions, drive 47 miles to US 89 near Big Water (a little west of Page, AZ). While a high clearance, 4 wheel drive vehicle is recommended, I've driven it in my RAV4.
You should check with the
BLM Visitor Center in Cannonville for road conditions - it is sometimes closed due to flooding or slides.
Another stop about 15 miles from Bryce is
Red Rock Canyon. Both camping & hiking opportunities.