We spent September 2022 in Crete using only public transportation (buses, ferries, taxis). This requires a little more advance planning, and you may not be able to see all the small sites, but you can easily fill 7-10 days.
We have not yet been to Rethymno but from what I’ve read, it might make sense as the first base for your family. We are starting our 2023 vacation there, with a transfer from Chania airport to our hotel in Rethymno. There is a bus, which is cheaper, but we find it worthwhile to splurge for a pleasant arrival in a strange place after a long trip.
We’ll spend a couple of days exploring the town on foot, and do day trips by bus, including one to the ruins at Phaestos and Gortys.
Then we’ll take a one hour bus to Plakias, where there’s a great beach and plenty of walking/hiking options. Lots of boat tours are available there as well. We visited Plakias last year and I think you & your kids would love it.
After Plakias, we’ll go on by ferry to more of the little coastal towns. You may want to take a bus to Heraklion and see Knossos. Or just bus to Chania or Heraklion airport & home.
Hotels on Crete are booking up quickly this year so I’d recommend finalizing your itinerary as quickly as possible.