@snoWYmonkey is definitely the person to ask. I have never skied Park City, Vail or Deer Valley, so I can't really compare, but I think you would be fine at JH. It is where I spend most of my time skiing and I do not in any way consider myself an expert skier, maybe high intermediate (but then women do tend do underrate themselves). I was a low intermediate (that's probably being generous) my first trip to JH, and loved it so much I traveled out here at least once a year until I moved to Idaho.
Anyways, I ski off all of the lifts at JH, and I am not terribly comfortable in bumps or deep powder (yet). Casper is a nice area for intermediates in addition to AV, and then Amphitheater off the Gondola. There are different ways to get around the mountain. I've also noticed that in an effort to make the terrain friendlier to intermediates/families, they are doing more grooming. I have not been over yet this year since only the AV lift was running until there is more snow (not sure if this has changed). The cat tracks are mostly used to get from one section of the mountain to another, there are plenty of blue and double blue trails. The only green trails are at the very bottom, off the Teewinot lift.
If you have a car, or take the shuttle, Targhee can make a nice change of pace for a day if you do indeed visit JH. It is a little more "friendlier" terrain, definitely less crowded, but still plenty of terrain to challenge everyone, just try to visit when it is not foggy.
I would recommend taking a lesson your first day, it can be a good way to be introduced to JH (if no one in your group has ever visited) with someone who knows the mountain and strategies for skiing the terrain. I can tell you that after skiing at JH, other places will look less steep for sure. Or, if you want, let me know when you visit, and I would be happy to ski with you and show you around some of the less steep areas. I'm happy just to ski!