For Menorca I would only suggest avoiding a few resorts (Cala en Porter, Cala'n Bosch, Cala en Forcat, Cala en Brut, Calan Blanes - these are quite built up and not particularly traditional. (Purely my opinion, don't want to offend anyone).
I would recommend you look at:
Son Bou: very pretty, great long white sandy beach, nice resort centre, very family friendly.
Cala Galdana: smaller resort, but even more beautiful, cliffs, greenery, sandy horseshoe bay, quite upmarket.
Santo Tomas: small, family friendly, great beach, good reputation for family holidays.
Some of the South Coast villages are gorgeous and very Spanish (Binissafuller, Binidali, Canutells) Punta Prima is very nice and was very quiet, however there is a huge hotel (Insotel Punta Prima) there now, which is a 5* AI thing. Not my cup of tea but has a good reputation.
Perhaps the most traditional place is Fornells, a gorgeous fishing village on the north coast, but the beach is a couple of km away in Playas de Fornells (mostly private rental apartments and villas via Owners Direct and similar). Very nice though and totally Menorcan.
For Majorca, it sounds as though you want the North Coast, places like Soller, Deia and Valldemossa, but I have no first hand experience to draw on.