Bryce is awesome. Some tips that aren't on the "main draw"  (although you'd be perfectly happy just staying in Bryce and hiking the popular trails):

-Do the mossy cave hike (it's outside of the park boundary, but on the maps). It's been a dry summer, but if there is still some moisture in the ground, the mossy cave is cool and the nearby waterfall on the red rocks is great.

-Lower Calf Creek Falls in Grand Staircase Escalante National Monument is also popular and is a hike within driving distance of Bryce.

-Swing through Escalante and visit the Hole in the Rock Museum and then go up over the Highway 12 Hogsback (worth the drive) into Boulder and visit Anasazi State Park. There is an alternate route to get to Bolder (see below for the advanced travelers.)

-Consider dropping south into Kodachrome State Park, another quick drive. 

Alternate Travel Plan to and from Bryce Canyon: Your GPS will take you to Bryce via Highway 12, this is good as you will drive through Red Canyon. To get home, if you can spare a few hours, continue east on highway 12, this will take you across the Murphy's Hogsback into Boulder (if you didn't do this earlier), highway 12 will then swing north and connect into Highway 24. Turn east here (don't go north to i70). This will take you straight through Capitol Reef National Park and the Hickman Bridge (1 mile hike) is right off the highway. Add another National Park to your bucket list if you can spare the extra 2-3 hours getting back to the SLC airport. Then just follow the GPS instructions to get back to i15. This would add a bit of time, but a great detour to show you various vistas, plateaus and another National Park of Utah at the expense of a few hours of planning. I can also provide route information on this as well.

For advanced adventurers:
-Ghost Town Tour (if your rental is an SUV and you are a bit more adventurous/advanced travelers). North of Bryce Canyon Intersection at highway 12 is a road called John's Valley. You can take it up to the Ghost Town of Widstoe which has a cool pioneer cemetery and some old buildings. That's all paved and rental car friendly and recommended for all. The advanced part: A well maintained forest service road heads east which will take you up into the mountains across the LAST horsedrawn US-mail route across a spine known as Hell's Backbone and into the town of Boulder Utah. In Boulder there is the Anasazi State Park, with native American Ruins (worth a visit above) and then you take one the most spectacular drives over the Hogsback of Highway 12 back to your camp. The entire route is easy, but I'd recommend an AWD SUV and make sure it's good weather (rain or snow will close the mountain) I can provide more detail if you are interested.

-Grosnover Arch:  This arch is located just outside of Kodachrome State Park, but it is off a dirt path.  Don't take a rental car, truck or SUV here.