We're planning a trip to Grand Canyon, Bryce, Zion. Any suggestions where to fly in and out? (Salt Lake City, Denver, Las Vegas?) Is there a route that is the most picturesque? We have 10- 13 days in the summer.

I believe Las Vegas might be the closest MAJOR airport to the Grand Canyon. (Flagstaff is the closest city, but I'm not sure who, if anyone, flies there.) And, especially since you plan to go on to Zion and Bryce, I would recommend going to the North Rim of the Grand Canyon. Among its advantages is that it's less crowded and less touristy.
After Zion and Bryce, you might want to continue east along the Utah-Arizona border and spend a day and night at Monument Valley and them go onto Mesa Verde just beyond the four corners in Colorado. Both are definitely worth a visit. From there, the closest airport is Durango, where United flies from.

My wife and I are planning a similar trip this spring, too. We are flying into Phoenix because it is fairly close to the GC and the air fare was the cheapest. Oak Creek Canyon near Sedona is beautiful and it is between Phoenix and the Canyon.
I would say go where the airfare and car rental combinations are the best.
I once considered flying to Grand Junction Colorado, but the rental car was much more expensive, so we settled on Denver. But the new airport in Denver extracts such a high tariff on rentals that it is not economical anymore.
Also in the area of Bryce Canyon there is also Cedar Breaks National Monument, and Capitol Reefs National Park.
On the eastern side of the state, I recommend Arches National Park, too.
Then to break up the red sandstone visits, you might drive north from Durango to Silverton and Ouray. Beautiful drive on a clear day.
I also suggest not being in Zion over a weekend, particularly a holiday weekend. Zion is close enough to population centers that it can become crowded. I made the mistake of going there on the Sunday before Labor Day, and it was difficult to find a parking place.

We did such a loop several years ago, in April--so the North Rim of the Grand Canyon wasn't yet accessible. At any rate, here's what we did (which of course we think is the proper way to do it!):

Flew to Phoenix and spent a few days. We were able to get tickets to a MLB game at Bank One Ballpark, which was great for my son. Then we drove to the Grand Canyon and spent several days on the south rim. From there we headed to Bryce for a night, then to Zion for a night. The two canyons are quite close together. Finally, we drove to Las Vegas, spent two nights, and flew home from there.

We'd do the same trip the same way again--except I'd cut Las Vegas shorter, as I really didn't like it after all the natural wonders we saw. Friends of ours did the same loop in the reverse way and loved it.

I hadn't realized Utah was so full of natural wonders, so I'd consider building in more Utah sightseeing. By the way, on the way to Bryce from the GC, after you pass into Utah, you'll come to a town called Kanab. There's only one traffic light, I think. Turn right at it, and on the left you'll find a great drive-in restaurant called The Junction. It's a throwback to the fifties, with waiters coming right to your car window. The food is great, and cheap; I seem to recall a grilled cheese for 85 cents! My kids still talk about it longingly.

At the risk of going on too long, I just want to mention one other thing. Our local newspaper happened to have an article on Bryce and Zion canyons last weekend. The writer mentioned how hot they can get in the summer, particularly Zion.

I hope all this helps, and I hope you have a great time however you plan it.

Try to go see Capitol Reef National Park. It's not to far past Bryce. The road there takes you through Escalante National Park and the Dixie Forest. This is one of the most scenic routes in America and Capitol Reef is as impressive as the Grand Canyon. By the way we started our trip in
Las Vegas and headed toward Zion first.

You really can't go wrong. My choice would be fly into Vegas and drive to the South Rim 1st. This way you can check out the Hoover Dam. Then proceed to Bryce and Last to Zion (Cedar Breaks on the way). This route lets you also check out Lake Powell and the Glen Canyon Dam on the way to Bryce. You also are better off entering Zion from the East as this route (through the tunnel) is more scenic. Then--you can leave Zion from the West and (I forget the major road), but and head back to Vegas from the North. I know this approach is great heading into the Zior River Valley and seeing St George Utah, then you have miles and miles and miles of desert until you hit Vegas. A little bit of everything.