I love Barcelona, and it has a great variety of sights, but it is a big, bustling city with a heavy tourist load. Many of its sights pretty much require pre-purchased (generally timed) entry tickets to avoid very long lines. Given your family's prefernce for smaller places, you may want to spend less time there than most of us would, but you still need to read about the sights and make the necessary arrangements if any of the following end up on your target list: La Sagrada Familia, Parc Guell, Camp Nou soccer experience (sells out early), Palau de la Musica Catalana (tour required), Picasso Museum (expect it to be badly crowded), Casa Mila/La Pedrera, and Casa Battlo. It is not a trivial experience to plan a short trip to Barcelona because you have to fit those sightseeing time-slots together like puzzle pieces without knowing how much time you'll want to spend at each sight or how long it will take you to get to the next one.
I consider La Sagrada Familia about as close to a must-see as any sight in Europe. There's a reason the other places are on the list, too. But there are three especially good things to do while in Barcelona that do not require advance-purchase of tickets: walk around the Barri Gotic (walking tours available from different sources), visit the Sant Pau modernista site (multiple buildings--needs time), and a day-trip to Montserrat (walking opportunities, and should be cooler than Barcelona).
I think your family would really enjoy Girona. The medieval historic district is large and atmospheric, the wall is walkable, there are two historic churches with English audioguides, and there's a good, moderate-sized museum. Girona is a short train trip from Barcelona. You could head there immediately upon arrival, buying your train tickets once you get to Barcelona so you're not guessing when you'll be ready to hop on a train, and spend your initial days there, recovering from jetlag. If you want to visit Figueres and Cadaques on day-trips, it will be quicker and a bit cheaper to do that from Girona than from Barcelona. You don't need a rental car for any of those destinations. The train will take you to Figueres, then there's a scenic bus ride to Cadaques.
I can't help with beaches.
I wonder about the possibility of renting a car for a few days and heading toward the Spanish Pyrenees. I haven't quite gotten there but did enjoy the colorful Cerdanya Valley town of Puigcerda and the small historic villages stretched out along the highway. The town of La Seu de Urgell is also lovely. Puigcerda is almost on the French border, and you can pick up the narrow-gauge Yellow Train into the French Pyrenees just over the border in Bourg-Madame. It takes you to the highly touristy but impressive fortified town of Villefranche-de-Conflent.
If "fortified" sounds enticing, but the distance from Barcelona seems a bit much, you might consider the small historic village of Besalu with its fortified bridge. You can get there by bus from Girrona or even Barcelona, but with a short car rental you could undoubtedly find some additional interesting stops.
