Welcome!

The only area you mention with which I'm familiar is the Tetons, and that was a long time ago. My favorite hike was Death Canyon (which, despite its name, is absolutely beautiful). However, I note that you'll be on the Idaho side, with which I'm not familiar. It should be less crowded than the national park, which can be a zoo, especially around Jenny and Jackson Lakes.

The reputation of City of Rocks is as a rock climbing venue rather than hiking. That area of Idaho is in the Snake River Plain, which can be really, really hot. I've driven through (on I-84) when daytime temps were 115*. On the other hand, the last time I drove through Idaho (on my way home from Wyoming), in mid-August, it was so cool that I didn't have to use the AC in my car! That is very unusual, though. If it gets too hot, head for Stanley and the Sawtooth Range instead.