The free Bryce Bus Shuttle also provides a twice-per-day tour of the Park. It is informative + gives you a very good orientation of the sites + views that are available. It starts by taking you 18 miles on the plateau to the end of the road, and then works backward. The tour lasts 3 to 4 hours.
The daily entrance fee is $25, unless you have a "Golden Pass" or "Access" card that gets you in free of charge. You can also purchase at the gate a pass that is good for a week, which might work for you since you plan on seeing Zion + the Grand Canyon too.
Have a great trip!