Corniglia or Fiesole with little children

Look at Viareggio and the Versilia coast. Plenty to do beach, Lucca, Pisa, mountains. Fly into Pisa and you're there. Corniglia is a poor choice for your stated goals.

Here's an article about Marina di Massa. Nice little semi-resort community. You'd be 45 min. to Pisa or Lucca. 30 min. to Lerici or Sarzana, 90 min to Firenze. Or you can walk the Via Vandelli and feel like you're back in Colorado (article on the same site).
http://www.apathtolunch.com/2011/07/marina-di-massa-real-italian-summer.html

Thanks for the reply and the info, Mike! We appreciate it!

When our boys were that age we took them to the Adriadic side for the lovely beaches, which have a very gentle slope into the water which is warm, and the sand is lovely. We were in the Emilia Romagna region, not far from Florence. I know this is a little "off the wall" perhaps, but with kids that age, I thought you may want to take a look at a few beach days -- and the beaches and towns on this side are very family friendly. We also went to San Leo and San Marino for day trips to the wonderful hill top fortresses and towns, which our boys loved. The towns of Riccione, Rimini, Ravenna, Catolica are resort towns that Italian families flock to, and are pretty unknown to most Americans.
We have family there, and if not for that we'd never have seen these areas.

Fiesole itself doesn't really offer a great deal of hiking and parks. There is a little park up in San Francesco from where you can see the view of Florence and another small park next to it, near the cemetery. When I was a teenager we used to hide in the cemetery at night and scare the smooching couples and the tourists. There is also a little park at the Roman Amphitheater. Otherwise there is a bigger natural forest on the way to the Cave di Maiano. That's the area where they filmed "Room with a View" when they go out to the country with the horse carriage (the driver of the horse buggy in the movie was actually a friend of mine, Luca, and that was his real job, taking tourists on horse carriages downtown Florence). That area is nice for hiking, lots of cypresses and olive trees. It's a very long walk from Fiesole itself though. That's where we used to take girls smooching on our vespa and motorbikes. Unfortunately lots of pervert voyeurs and exhibitionists as well. I hated those people, especially because in those years there was the Monster of Florence to worry about.

For a beach for small ones, I agree that the Versilia area is a better option, compared to the cliffs at the Cinque Terre. Monterosso would be better than Corniglia, but even there waters get deep quickly. Anyplace in Versilia is good. I used to spend my summers as a teen ager at Marina di Massa, Marina di Carrara, Forte dei Marmi, Marina di Pietrasanta, Viareggio (depending on where we rented the beach house each year). From there we would take day trips to the Cinque Terre also sometimes (just 45 min away by motorbike). Viareggio is bigger and more lively (i.e. crowded) with the 'passeggiata' on the Lungomare (promenade). Forte dei Marmi is where the richer people go. The beaches anywhere from Bocca di Magra all the way down the Tuscan coast to Livorno are big, sandy, and don't drop too quickly, therefore perfect for kids. South of Viareggio, between Viareggio and Marina di Torre del Lago (Giacomo Puccini's birthplace) there is a huge pine forest. you walk through that to get to the beach, which is really nice and not crowded in that area. Viareggio and Torre del Lago are connected by a boulevard called Viale dei Tigli, that runs through the pine forest. We used to go there by motorbike, but I'm sure there are buses if you don't have your own wheels. More to the south there is also Marina di Pisa. North of it there is the Parco of San Rossore. Never been inside, but I heard it's a nice beach. I don't know if it's open since there is a NATO base there (Camp Darby). But I know lots of US servicemen and their families from the base use that beach. Camp Darby is considered the vacation base for US military personnel.