Considering you have posted this question under the Spain section, I assume you would like to see more of Spain, beyond Barcelona, before making a b-line to France/Italy etc.?
If that is the case Spain offers plenty in the way of history, parks, science, military museums, beaches, etc. Why not start in Madrid and make a road trip to Barcelona? Along the way you will find incredibly preserved Roman ruins, Crusader Castles, Royal Monasteries, a medieval town hanging on a cliff, and the most beautiful village in Spain. I'd recommend this route:
Fly in to Madrid
Madrid (3 nights) - See Spanish Naval Museum, rent car
drive south to
Toledo (2 nights) - See Spanish Army Museum
drive east, stop at the Roman city of Segobriga, continue to
Cuenca (2 nights) - stay at the Parador, hiking daytrip to Ciudad Encantada
continue east until
Valencia (3 nights) - see City of Sciences, day trip to Albarracín (prettiest village in Spain)
drive north along coast, stop at the Knights Templar Castle at Peñíscola, continue north
Tarragona (2 nights) - see Roman ruins, day trip to Poblet Royal Monastery (UNESCO Site)
continue north to
Barcelona (4 nights) - drop off car
Fly out of Barcelona
I think you and your family will have a much more rewarding time if you use those 2.5 weeks to focus in on Spain, considering you have already been to France and Italy.