Arriving in Barcelona mid-June. Need to choose departure city, not necessarily in Spain.

If you are not too fond of the miserable summer heat you are wise not to venture further south than Madrid. Spain has plenty to offer up north, where it is much cooler and more green and lush. Have you been to Barcelona or Spain before? May I also ask how many nights you will have available?

We will have up to three weeks. We would love to explore the hotter parts of Spain in cooler weather another time. We’re open to exploring other countries in the area as well during this trip.

I was in Madrid for a few nights 25 years ago. Spain will be completely new for the rest if my family. We’re all excited to explore new places together.

You could take the fast train from Barcelona (6.5 hours) to Paris that topped out at 184 mph when we were on it in June.
We flew into Madrid and loved the city. It'd be a good pick for a secondary city.
And as you mention, budget European airlines can take you virtually anywhere you like.

We like to visit 2-3 cities in close proximity to each other and then take a budget airline to the other end of Europe--to a completely different type of place. Somewhere like Copenhagen.
If you're budget minded, going east is much less expensive--like Poland or Hungary.

The north is Pyrenees, Basque, Galicia. Good with a car, slower with public transportation. You could easily spend a week or more and you'll want at least 5 nights in Barcelona.

Cheaper destinations that spring to mind that probably won't be hot are Ireland, northern England and Scotland, not so good for driving ("wrong" side of the road, many roundabouts and lots of narrow roads lined with stone walls but no shoulders). Dublin has just about everything you've listed and day tours to beautiful countryside locations and there are trains/buses to smaller cities. I haven't been to the others. You may get a good price on flights home from Dublin.

Another possibility is the Netherlands - Amsterdam is great, and it's easy to day trip through the country with its excellent rail system.

Well, as a Spaniard I am partial that you stay and explore my country ;-)

If you have not been to Barcelona yet, I recommend at least 5 nights, that way you can see the city properly and squeeze in a day trip or two. From Barcelona, there are a number of unique day trips! From Barcelona I'd take the AVE High speed train north west, to Zaragoza, after a few days there rent a car and road trip westwards across Northern Spain.

June/July is the perfect time to explore Northern Spain aka "Green Spain". Very different from the rest of Spain, feels like a mix of Switzerland and Ireland, especially the Picos de Europa region. With a car, you can start in Zaragoza, drive up to San Sebastian/Bilbao and drive west along the emerald coast of the Bay of Biscay, then up into the jagged mountains of Asturias and Cantabria, finishing in the misty forests of Galicia (a Celtic Culture), which has some of the best seafood and white wines in Europe. You can finish at the famous pilgrimage town of Santiago de Compostela, in Galicia, then drop off the car and take the Alvia High speed train to Madrid, spend a few days, then fly out from there.

Are you or your boys interested in cycling? Depending on when you go, you could catch the Tour de France. They hit the Pyrenees on July 4th and 5th.

Thanks so much to everyone who has posted. I appreciate your input.