These two parks are 85 miles apart and look like different planets. Zion is a deep river canyon: the Virgin River spent millions of years cutting through Navajo sandstone until the walls stood 2,640 feet above the floor, and you see it from the bottom, with cottonwoods and water. Bryce is the opposite. Bryce Canyon isn't a canyon at all, but a series of amphitheaters eaten into the edge of a plateau at 8,000 to 9,000 feet, filled with thousands of orange and pink rock spires called hoodoos. You look down on it from the rim.
Five days is the right length for both. Zion needs two full days because the best things (the Narrows, Angels Landing, the Emerald Pools) are far apart and the shuttle sets the pace. Bryce needs one very early morning and one more day for the long drive to Rainbow Point and a night under what is one of the darkest skies in the lower 48. The price of the combination is driving: this is a trip with a rental car, 85 miles between the parks and 160 to 260 miles from the nearest big airport. That is why the plan has a transfer day, with a stop at Kolob Canyons.
The days: Zion Canyon by shuttle, then Scout Lookout and the east side, Kolob Canyons and the drive to Bryce, a full Bryce day from sunrise to the stars, and Highway 12 home. Start on a Monday. The parks never close, but both fill from Friday to Sunday, and this order puts your Zion days on Monday and Tuesday, Bryce on Thursday and the long drive home on a Friday. The plan works in any season, with winter notes in the text. For the country as a whole, see the United States page; for the cities you'll probably fly into, see 3 Days in Las Vegas and 5 Days in Salt Lake City.