The tuweep area (permits required) is a no trailers area.. also, quite remote and they have the worst road conditions in 10 yrs, quite beautiful though. Whitmore canyon is also very popular, remote, and really bad roads right now as well. The "North Rim" national park and Grand Canyon highway are closed until memorial day. the best viewpoint on pavement is likely to be Horseshoe Bend near Page,AZ from above and Lee's ferry from down inside, unless you make the trek to the South rim.
Expect to see snow and have chili nights while you're at Bryce Canyon.. but soon as you drop down to Zion things should get much warmer. If I were going from Bryce Canyon to Zion and then South to the Grand Canyon I would leave Bryce and go down highway 89 until the highway 14 junction down cedar Canyon then go down I-15 from cedar city.. down to Toquerville and then over to Zion. I would then drive through Zion back to highway 89 and go south through Kanab toward the grand canyon. That is the all payment play it safe route anyway.. with no backtracking and beautiful views the whole way. A lot of the backcountry routes are still going to have snow on them there was six to eight feet in a lot of areas when I was up there last week.