Susan and Monte, flying into Las Vegas (much as I hate that city) might make better sense. In a perfect world - if you didn't have your accommodations set already - I would have done LV>Zion>Bryce>Page> NORTH RIM of the Grand Canyon and back to L.V. As you could have hit Zion earlier the day from there (shorter drive) than Page from Phoenix. The North Rim would also be closer to Page than Zion from the South Rim.
Yep, those mule trips to the bottom of the canyon from the South Rim are all overnighters and sell out very far in advance. They're also no picnic to do depending on how comfortable greenhorns are for hours in a saddle!
OK, easy hikes:
Page is a no-brainer: Antelope and Horseshoe. It may be very hot or at least uncomfortably warm at both. Horseshoe is not a long hike but a lot of it was a slog through sand. There really isn't much else to do there with the 1 day that you have. Not our favorite part of the Southwest, and we prefer doing our slots without a tour but I know Antelope appeals to many.
Bryce: The Queens Garden/ Navajo Loop is the classic hike there. We've done it from both directions and I don't know as it makes a huge difference which one you choose. You may start the loop at either the Queens (Sunrise Point) or Navajo (Sunset Point) trailheads although the latter is usually recommended for a tad easier puff back up to the rim. It's about a 3-miler, RT.
Otherwise, check out the various points (Fairyland, Bryce, Inspiration, etc) with your remaining time. Best time for photography from the points is earlier or later in the day: the landscape flattens and loses depth in overhead sun.
Zion: Definitely do the bottom-up Narrows day hike if conditions allow (check with rangers). I would recommend renting the footwear package from Zion Adventure Company in Springdale, and do it the day you get into town as their rentals are popular.
https://www.zionadventures.com
https://www.zionadventures.com/narrows-rentals-471276.html
https://www.zionadventures.com/plan-your-narrows-hike.html
The shoes will help you keep your footing on slippery rocks, and poles help you stay upright in swift current + test depth of pools in front of you. Neoprene socks insulate your feet during the hours they will be in the river. Wear clothes that can dry quickly: no heavy cottons.
And keep an eye on the Zion website for trail closures? Some are currently closed "until further notice" due to storm damage. Lower and Middle Emerald Pools are open but not Upper, and Kayenta Trail is closed as well, which is an easy but very pretty trail. All of Kolab Canyon is closed to construction.
https://www.nps.gov/zion/index.htm
South Rim:
The easiest is the 13 miles of rim trail: 8 miles from the Village west to Hermits Rest, and 5 from Village to South Kaibab trailhead,
https://www.nps.gov/grca/planyourvisit/upload/RimTrail.pdf
There are some shuttle points along the way (no private vehicles are allowed on the road to Hermit's Rest most of the year) if wanting to catch a ride here and there but be aware that some of them along Hermit Road don't stop at every point on the return trip from Hermits Rest:
https://www.nps.gov/grca/planyourvisit/shuttle-buses.htm
The trail through woods between Park Headquarters and the rim is not dramatic but pleasant (see map linked) and the Pioneer Cemetery (near Shrine of Ages/Headquarters) is an interesting ramble.
https://www.nps.gov/grca/learn/news/upload/sr-pocket-map.pdf
http://www.grcahistory.org/sites_southrim_cemetery.html
Otherwise, you might hike a bit under the rim on South Kaibab or Bright Angel: just remember that every step DOWN is another back UP! And check the schedule for free ranger walks/talks or evening programs: always free and usually interesting! This is the current schedule but will change for fall:
https://www.nps.gov/grca/planyourvisit/sr-programs.htm