You'll have a blast no matter what you do. Too many good options to list. The best ones come down to preference. That said, here's what I would do if had your schedule and two friends who were fit, adventurous hikers but desert neophytes (one bullet per day, double bullets are primo food options in each area).
-Fly into Vegas, rent car, buy groceries, drive to Springdale. If early flight, hike Angels Landing, watch the sunrise, picnic dinner on top, hike down via headlamp. Car camp at Coal Pits Wash TH (free). If late flight, do Angels the next morning.
--Mean Bean for coffee, Pizza Noodle for dinner.
-Drive to Bryce. Hike Fairyland, Navajo, Peakaboo trails. Camp north of park on FS land (free).
--breakfast at Bryce Pines is good. Do NOT eat at Ruby's Inn.
-Drive to Escalante. Get water and beta. Drive to and hike Peakaboo/Spooky. Drive to Egypt TH. Camp.
-3 day backpack. Egypt TH to Fence Canyon, explore Neon and Ringtail canyons, hike out 25 mile Wash and cross country back to TH. Get beta from Steve Allen's books and bring good detailed topo maps.
--pizza, beer and ice cream at Escalante Outfitters. You could reserve a cabin, too.
-Drive to Buckskin Gulch. Shortcut down the Pariah. Mind the washboard. Out and back exploration from Wire Pass TH. Camp (free) at TH.
-Drive to North Rim Grand Canyon. Dayhike down North Kaibab to hearts content. Camp outside park at Parrisawampitts Point (free, bring water, beer).
--breakfast and cookies at Jacob's Lake Lodge are good. Beer from Teddy Roosevelt bar on North Rim lodge porch, too.