You generally need to apply for forest service permission 6 months to a year in advance to hike down and spend the night in the Canyon. And you can not sneak down there. Day hikes are always permitted except when there's lots of snow on the trails. Summer is not the best time to visit, might be 80F on the rims, but 95F in the canyon. I wouldn't bother with a canyon trip from mid-June to mid-Sept. Too hot, way too crowded. They get 6 million or so visitor per year, most in the summer.
Since you didn't give dates, but just say 'summer', I'd recommend you just drift north and save the desert southwest and the GC for some other year when you can make the trip in the spring or fall. This trip just try to visit mountain areas. There's lots of those too. Like most of Montana. But the most comfortable areas in summer tend to be up north.
Give us some dates for better advice.