Easy hikes through the general caverns.
I would suggest hiking in through the natural entrance, completing the Big Room, and then going up though the elevator for lunch (we didn't have any wait time). We packed our lunches, but they have a restaurant and gift shop on site. After, I would suggest making a reservation to go back down for any tours- FYI: King's Palace is the only one you can take a 4 year old on.
-Make reservations online as they fill up pretty quickly. I made mine online and then ran into truck problems and they were nice enough to reschedule us into open spots. The people there were very nice and accommodating, the people on the phone that I called to let them know what was going on- not so much. If you happen to have problems, get in touch with the actual people who work at that location, not the general national park services (that number was listed online).-
Anyway, after lunch, we left through the natural entrance to catch the bats leaving the caves, but it was moderate-strenuous. There was a woman who worked there that warned me that it was a strenuous 80 story ascent (probably because I had small children), but we were able to complete it. Just be aware there are a lot of switchbacks and it's a steep incline. But it was so beautiful and I definitely wouldn't skip out on this by taking the elevator both ways.
The temperature in the caves is 56° pretty much all the time, according to the rangers. We went in toward the end of June and it was cold enough that all my kids wanted sweaters, even with it being hot outside. So be prepared. In the regular caverns, you won't really need any flashlights. They have lights on pathways. But I read that if you do the lower caves (can't take children here), you will need them.