Transatlantic prices for this summer are still pretty crazy right now, so you may wish to wait.

Since MEM is not the most competitive market, it can be economical to buy a positioning flight to a gateway with a more competitive market (e.g. BOS or JFK) then purchase your transatlantic flights from there. This is harder on a family vacation, as AA will make you reclaim and recheck your bags, even if the positioning flight is on AA.

A multi-city MEM-BCN / BCN-NAP / FCO-MEM itinerary all on one ticket will probably price as more expensive than buying MEM-BCN / FCO-MEM and purchasing the BCN-NAP flight separately, because AA transatlantic fares usually charge a decent amount for a stopover or combinability.

The Italian airports, especially Rome and Milan, seem to be some of the cheapest to fly to in Europe right now.