Traveling to Spain for 10 days in June. What other cities should I see

What stands out to you--what kinds of things do you like to do? Will you stick to train or rent a car?
https://www.spain.info/en/

Ideally we would travel by train. Could take short flight back to Barcelona to save time.

That probably will not be necessary, as the train system is excellent. Have you considered the most obvious, Madrid?
When I am planning something like this, I consult a guide book first, but I also look for a map of the train system to make sure I select things I can easily get to (when using public transport). Like this:
https://www.mappingeurope.com/spain/spain-rail-map.htm
Now you know what's easy to get to and you can read to see which ones are of interest to you.

I spent 4 nights in Barcelona recently and it was not enough. It's a great city, lots to see and do.

Might you consider adding one or two more days in Barcelona?

If you don’t have a guide book yet
https://www.ricksteves.com/europe/spain

Valencia should not be overlooked. It's (so far) my favorite city in Spain.

That’s very subjective. Part of the answer lies is how you’re getting around. Having visited Spain numerous times, I’d rent a car and visit any number of small towns and villages. If it’s your first trip you’ll probably want to stay with the standard cities tourists that are easy to reach via high speed train. From Barcelona, maybe Alicante, Tarragona, Valencia, or Zaragoza.

Because we do not know anything about you, we cannot be your travel agents and create a "dream" itinerary for you. This kind of question also shows that you probably have not done a lot of advance research on your own, which is why people keep telling you to consult a guidebook.
It is likely that you have never been to Spain before, so why do you want to go?
What do you want to see and do?
For me, Barcelona is great for: Gaudí architectural creations, big cruise port, nice beaches. It has also become very busy with tourists, and last year they had a small drought. The famous religious monastery/sanctuary of Montserrat is nearby.
But, I have many reasons why I travel to Spain, and I have many, many places to visit.
Do you like history, castles, cathedrals, hiking, wilderness, Moorish architecture (and/or Mudéjar), beaches, resorts, tapas and late dining, wine, flamenco dancing, etc., etc. Sometimes I just want to hear some "puro Castellano."
Also, another key issue is budget--are you trying to save money, or willing to splurge?
So, as a long-time visitor to Spain, I can say that I will rent a car to do a unique itinerary (stay in a remote parador) or just take the train when I am going from city to city (Madrid to Barcelona).
If you have high energy and a good budget, you could fly/travel to a few different places, but the week will go by quickly. You could consider Madrid and surroundings (Toledo, Segovia), or Sevilla and south (Granada, Málaga, Córdoba), or Valencia and/or Mallorca, or Basque area, and so forth.
Each person who responds could give you a different recommendation, so I always tell first-time travelers to look at a tour company itinerary and get an idea of what they highlight. You can certainly find a lot of information from the Rick Steves' guidebooks, tours and videos! ¡Buen viaje!