The Jenny Lake hike is a nice one. A few miles east of the park boundary on Hwy. 26 is Turpin Meadows which has some trails that aren't too tough. Carry bear spray though!
Table Rock from the Idaho side is a great hike that is around 13 miles round trip and 4,000 vertical feet and the views are outstanding. The first time I made the hike I was 12 years old, but if your kids are too much younger than that I probably wouldn’t recommend.
Edit: just realized you said June and not July. There’d probably be too much snow that time of year for the hike. Be bear aware and packing firearm/spray would be good.
Can't help you with kid hikes but like wyosteve said "carry bear spray" (and/or a firearm). Don't be like the hikers I see around the Big Sky MT area with no spray or situational awareness.
Take the ferry across Jenny Lake, and hike as far up cascade canyon as you want/can. Then, turn around and head down. Maybe take the trail back to the parking lot vs the ferry.
I agree with the posts above. Take the ferry across Jenny Lake and hike as far as they want to go. Inspiration Point is not too far and there are a couple of waterfalls along the way. I think they would enjoy the ferry ride as well. We did it last summer and it was pretty cool.
The high alpine will likely still be snow covered in early June, just fyi. Cascade canyon gets better the further up it you go, but again, the higher you go, the more snow you’ll run into.