Sorry for the long post, but I used to live in the San Francisco area, currently live 60 miles from Yosemite, and have been on road trips to Utah/Arizona National Parks 4 (5?) times in the past few years. I'd say 20 days for the parks you've mentioned is easily doable, if you're willing to spend a little more time driving to see more and stick to highlights. As I recall, two of my longer trips were around 14-16 days and mainly included Zion, Bryce, Capitol Reef, Canyonlands, Arches, Monument Valley, Page, Coyote Buttes. A couple of spots were only on one trip, not both. Of course, many people would rather take it slower and spend more days in fewer locations to explore more thoroughly. Plus, most of the spots you mention also have a lot of great locations nearby that may be lesser-known, taking up more time at fewer locations.
I don't recognize the magazine/newspaper you mention, but I'd highly recommend Photographing the Southwest: Volume 1--Southern Utah (2nd Edition)
. Lots of good advice on the state, including photography-specific tips and tricks, and both popular and lesser-known spots. The Arizona one is also great, but maybe not worth it as you may only hit the Grand Canyon there. I was not impressed with the Northern California book (different author).
I probably have some good bookmarks on my home computer, but right now I'm on the road for work. If you read German, or are willing to use an online translator, you may find this site informative. http://www.synnatschke.de/
The wife & husband photographers have visited the SW & California and have travelogues, describing their experiences in some of the locations, including directions, tips and topo maps for some spots. They also have other sites with mainly just pictures (googling their last name brings them up).
Most the National Parks charge $15-25 entrance fees. When you get to Yosemite (or whichever is first one you enter), get the America the Beautiful
pass. It's $80, and gets you into all the National Park for a year. And if you're arriving in less than a month, I'd be a bit concerned about booking hotels near these places. You're entering the tourist season and some of these spots (like Bryce) have limited options nearby, unless you'll be camping/sleeping in your car.
Off the top of my head, here's a list of some of my personal favorite spots or ones I'd highly recommend on a potential travel route. Well, at least these would be highlights, I could definitely recommend more but you wouldn't have the time, even at a brisk pace. I also have dozens of spots in Utah/Arizona on a google maps custom map I've used for route planning; if want a link to it, send me a pm.
SF down to Monterey/Carmel/Big Sur/Point Lobos area.
Yosemite (Personally, I'd skip Sequoia, but that's just me)
Down 395 (great views most of the way) to Death Valley/Joshua Tree
Through Vegas to Zion (I'd recommend swinging through Valley of Fire just north of Vegas)
Scenic byway 12 on the way to Capitol Reef, possible quick stops at Devil's Garden (not to be confused with the Devil's Garden in Arches) & Goblin Valley.
Canyonlands & Arches
Page (Upper & Lower Antelope Canyon; you'll be able to see the popular beams, but I recommend the more expensive photography tours if you go; Horseshoe Bend).
Grand Canyon (note that the normal road between Page and the Grand Canyon has been closed for months due to a landslide, as was closed in early March when I was there. I'm not sure if it will be open by May; if not, factor alternative routes in your plans).
I'm not sure if you're flying home from SF, or some other location like Vegas, so wasn't sure where to go after that.