I am flying to Barcelona on July 3 with my wife and children (ages 10 and 12). After three days in Barcelona I will pick up a rental car.

Any suggestions for places to go/things to see would be appreciated greatly.

We're travellers and want to see as much as possible. Our flight back is on July 16.

Thanks in advance!

You might want to stop for a day in Pamplona. San Fermin is Jul 7 and the festivities (day & night) continue for a week.
You might want to check out resorts along the Costa Brava and Costa Dorada (close to Barcelona) or the north coast from San Sebastian westward to Bilbao and all the way to Asturias. (Santander, Laredo, Comillas, Santillana del Mar, San vicente de la Barquera). July is prety busy there

Joey, I would not go to the san fermines in pamplona with children.