Skiathos town is fine for kids, you just need to be mindfull that as it is a pedestrian zone with all cobbled lanes the access for most is by scooter.
It does not get rowdy and the clubs are only open peak season and are situated away towards the airport.
Its a laid back kind of place but you can find somewhere more lively if you need to.
Plenty of boat trips from the old harbour, either to Arkos, Llaria, Tsougria, round the island or over to Skopolos, and Alonissos and the marine park to see the dolphins.
Beaches are best accesed by bus, and every bus stop from no 5 onwards is near or by one, you cannot go wrong, but if you like the freedom of a car its up to you.