Been to both rims many a time along with Zion, Bryce, Escalante and many places in between. My observations:
The south rim is more crowded, but you are able to drive in a more continuous manner along the rim. There's a lot of different view points on the north rim, but several of these require a fair amount of driving to get to. The south rim has more amenities. One that is really nice is the shuttle system. So once you park, you can visit much of the south rim w/o moving your vehicle. I think photography is easier for us novices from the south rim. Of course this doesn't matter as much in the summer as the sun is virtually overhead, but in certain times of the year, shooting toward the south often is difficult with the sun in front of you. North rim is cooler due to elevation. While Mather is a nicer CG, Trailer Village is the place to be if you "need" a/c. Of course if you're out exploring all day, it won't matter as much since it tends to cool off quite nicely in the evening. That's the great thing about a PUP. Unzip every window and let the breeze blow through.
As for the suggestion to hike into the canyon, I'd only do it if you got some cooler weather. Gets quite hot as you go down. If you do want to hike, take the Bright Angel Trail and leave in the mid to late afternoon. The shadow from the canyon rim hits the upper part of that trail so you're out of the sun. Plus, when you get back to the top, you can get an ice cream cone at the Angel Lodge!
The north rim is simply too far from the other places you're talking about visiting to use it as a base camp.
All told, if you haven't been to the GC, I'd recommend the south rim for your first visit.
For Zion NP, I'd suggest getting a reservation at Watchman CG in the park. Electric service is available plus it's walking distance to the visitors center where you catch the shuttle bus into the canyon (the only way you can get up there). Watchman also borders the Virgin River which flows pretty slow by mid summer. Kids build little dams and play in the "river" (really more of a stream) and the adults set their lawn chairs up in the water under the shade of the giant Cottonwood trees. Great way to spend a hot part of the day and way better than a motel swimming pool IMHO. Otherwise, hike the trails on the east side of the canyon in the morning when they're in the shade and the west side trails in the afternoon when they're in the shade. Do plan on a day to hike up the Zion Narrows. You end up wading up the stream with walls that are over 1000' high and a canyon that is just a few dozen feet wide in places. But it's much cooler in there, even in the middle of the day. Bring water shoes or tennis shoes that you don't mind getting wet and sandy.