I would love to help but my information is way out of date as it is a few years ago since we have been, in fact I mention Pal Beach in my post Forgotten Palamos. If the place hasn't changed, which is a possibility, then you would struggle on a 14 night holiday with 2 young ones, as beautiful as I remember the place to be, I would advise car hire.
We would walk along the coast, slowly, which had two purposes, one was to see a little more of the area and the other was to tire out the children LOL. At the time we were in Palamos we had friends, with children, staying not far away at Playa de Aro so we would go and visit, taking the bus.
So as I say, apologies for this post as its not really of much use, I appreciate that, but on recollection, car hire would be advisable.