IMHO, the best are trail runners, which have a nice gripping sole (Vibram or similar) and are supportive. However, for kids, you're probably looking at a running-shoe style with a lug sole that will grip the ground (or rocks) instead of sliding. Not necessarily Vibram, but similar. You do not want waterproof shoes (or Goretex-lined), because their feet will sweat, turning waterproof shoes into a sauna. Many of those areas you're visiting are going to be hot. Shoes with really good ventilation will dry faster and allow the feet to breathe, but also let in more dirt.

You should also get several pair of either merino wool or breathable synthetic socks. No cotton, because they will absorb perspiration, become soggy and cause blisters. I personally prefer merino wool, because it feels dry to my feet even if not completely dry, but that's an individual matter, known in hiking circles as Your Mileage May Vary.

Of course, good fit is vitally important!

Whether a $60 pair of shoes will hold up under the stress of hiking and then clean up enough to be suitable for school wear in the fall is another question. They are going to get pretty grubby. After being on a recent shoe shopping trip with my 11-year-old grandson and his mom, it appears that $60 is very low-end if you want the shoes to last. I'd be inclined to get cheap shoes--if you can find somewith lug soles and that fit--and be prepared to toss them after your trip.