Bryce and Zion
 

I am planning a quick trip from Salt Lake City to Bryce and the next day to Zion. I only have 1 day in each park. Where should I stay and which trails are “must sees”

Bryce: Early in the day, drive to the various look out points on rt 63. Inspiration, Natural Bridge, etc, all the way out to Rainbow Point. Hike the short Bristlecone trail.

Mid day, hike the Queens Garden/Navajo loop figure 8 trail.

End of day, drive to Zion and stay in Springdale.

Zion. As of this writing the shuttle system is not operating so they are only letting ~400 cars per day into the park proper.
https://www.nps.gov/zion/planyourvis...t-my-visit.htm

Demand is likely to be many times that so expect to be turned away. The link above give more info (for mid June). Summertime it will get worse. If getting in by car is a must-have, then you'd need to be in queue and ready to go very early in the day. I have no idea if that means lined up a 4am, 5am, or whenever so hopefully someone else can chime in with exact current conditions.

Even if you get in with your car I would not count on being able to drive between the various trail heads. If you bring a bike you can navigate the entire canyon without the car limits. The canyon is ~10mi end to end with a gradual uphill all the way. Very rideable if you are a regular bike rider.

Trails: If you get in either by car or bike, start at the far end at the Temple of Sinawava & hike a short distance up the river into the Narrows. Even a hike just ~1/2mi past the end of the paved Riverside Walk trail will give you a good feel of this iconic hike.

Other hikes - my favorites are Scouts Lookout (on the approach to Angel's Landing) and Observation Point. The former will give you spectacular views of the canyon 1000' below and you can see hikers traversing the narrow ridge to get out to Angel's landing. Or go for it yourself.... Both of those are listed as "strenuous" hikes but well worth it in my opinion.

If you are not game for Scouts Lookout, then you can do the less strenuous Weeping Rocks & Emerald Pools trails. With limited parking you'll need to map out your parking day. If you brought a bike and can get in by car, then consider parking for the day at Zion Lodge then riding around from there.

Good advice from J62. If it is too crowded at the valley, look to the separate section for a hike or head up to Cedar Breaks which will probably have fewer visitors.

What time of year?? That makes a big difference.

Zion just announced that as of 7/1 cars will be banned and shuttles will resumed. Reservations required.
https://www.nps.gov/zion/index.htm

Bryce and Zion
 

Bryce - almost the only place to stay near Bryce is Ruby's Inn. It's a fun place though with lots to do (horseback riding, ATVs, etc.) Also, if you want to splurge on a dinner show, Ebeneezer's is great. My favorite hike is the Navajo Loop/Queens Garden trail. It immerses you in the canyon and has a lot of unique hoodoos, tunnels, and rock formations.

Zion - all of the hotels in Springdale are pricey. If you stay in Virgin, Hurricane, or St. George you will find lots of cheaper options, though they will require more driving. Hiking in Zion takes a long time because you have to find somewhere to park, ride the shuttle, do the hike, ride the shuttle back, find your car, etc. One of my favorite hikes is the Canyon Overlook trail. It's got some amazing views and some edge-of-the-cliff hiking (with guard rails) that's really cool. Plus, you can drive your car right to the trailhead without riding the shuttles. You also get to go through the Zion-Mount Carmel Tunnel on the way, which is amazing.

If you only have one day at each, you may need to just spend the night in between the two parks in Springdale. There are many hotel options, I like the La Quinta Inn & Stuies if you can get a good rate, worth checking the different third-party sites like hotels.com, Priceline, etc. I think it may still include some kind of breakfast. The rooms have refrigerators and microwaves.