Hey, I just registered on this site but already live in Monaco. I'm Dutch and my fiance (English) is just joining me in Monaco now. We just had a little boy and so we are also looking at what we can do with him. Well, only a short drive away are the mountains, so during the winter you can enjoy some snow sports and in the summer, picknicks and hikes. In the summer time, the beach in Monaco may be packed but there's nothing against taking a little drive out. Plenty of coast line to pick from. I don't know how old your boys are but there are some playgrounds in the little parks of Monaco where they can play (if they are not too old).
Let me know how you get on, good luck with the move!
AW