Great stuff about Big Bend NP.
Update:
We went to the North Rim of the Grand Canyon and it was fantastic. We got there early, before 8 AM local time, mainly because AZ doesn't observe daylight savings time so we gained an hour on our trip. We went to the major stops along the way and generally were the only people there. Even later in the day the crowds were sparse. In fact, the only place we saw a large number of people was at the visitor center but I wouldn't call it overcrowded. Not much hiking here (just a few short walks). It's just too scary worrying about my kids. That's my story and I'm sticking to it.
The next day we hit Bryce in the morning to early afternoon. I loved this park. We got there around 8:30 and were able to find parking fairly easily. Everyone said to be there before 10 and that was good advice. Once we were in the park and parked we were able to take the trails and use the shuttles without much of a problem at all. There were a lot of people there and we certainly weren't alone on the trails but it wasn't annoying. If a big group came by we would just wait for them to pass and continue on. You can put me on Team Bryce. I enjoyed my time there more than Zion.
Wait, but I didn't get to see much of Zion, right? Well au contraire, mes amis. On advice from articles and a tip from a friend we went back to Zion in the afternoon. We got there at about 2:30 and sure enough, there were no lines to get on the shuttle or anywhere else in the park. I liked Zion well enough and it is a unique desert river environment but it seems to really get the Holy Crap! experience you need to go to Angel's Landing and, well, I'm just too old for that crap anymore. I'm ok climbing mountains and walking a few miles but 1500 ft climb and 5.5 miles round trip is too much for me. Congrats if you can do it though, it sounds terrifying at the top. The part of Zion I actually like the most is the stuff on the Eastern side of the tunnel. I think it's the same reason I like Bryce better as well. The science nerd in me loves looking at the geology of the region and I just find the sandstone formations on the Eastern side of the park more interesting. Just like Iooking at all the hoodoos in Bryce. Bottom line, if you want to visit Zion, either get there before 6AM or after 2 PM. Fortunately it was pretty cloudy for us so the temperature didn't get much above 90 for us. Zion can be really hot in the afternoon (hence why the crowds go down).