Here's a little info about cycling the Grand Canyon's South Rim. The elevation is about 7,000 feet, so if you'd be coming directly from Florida be prepared for the altitude change. The road along the South Rim is about 30 miles long, so a rim tour is not a long one. From Grand Canyon Village west to Hermit's Rest (about 9 miles along the rim), only shuttle busses and bikes are allowed durng the summer. There is a multi-use trail along a relative short section of the rim. There are 3 routes to the South Rim. US-180 and AZ-64 from Flagstaff is about 80 miles, initially climbing, then decending to Valle, then climbing to the rim. US-64 from Williams is 60 miles, decending gradually to Valle, then climbing to the rim. AZ-64 from Cameron is 32 miles of serious climbing -- about 3,000 feet. Except in the Flagstaff and rim areas, there is little vegetation. Dogs are not permitted on trails in the park.
Thunderstorms during the July and August monsoon are usually in the afternoon, very scattered and generally of short duration. They can be quite violent with lots of lightning.
Since the South Rim provides such a short ride, you might consider riding other routes in the vicinity. Old Route 66 west from Flagstaff to the California border is a great tour. The original road can be ridden except for 40 miles between Williams and just west of Ash Fork, where riding the I-40 shoulder (legal in Arizona) in required. Another nice tour is riding southeast from Flagstaff on Lake Mary Road to AZ-87, west on AZ-87 and AZ-260 through Camp Verde to Cottonwood, then northeast on US-89 to Sedona. US-89 south from Flagstaff through Sedona to Cottonwood includes a very steep and winding decent down Oak Creek Canyon.