Need some advanced itinerary guidance

Any chance of renting a car? That'd be far easier and less rushed for you, otherwise I am afraid you're going to have a rather complex time to set up the itinerary and if you're on a "tight" schedule you're far too dependent then. Moving about in rural areas is all right but schedules might not always adjust to a busy itinerary -if you catch my drift.

I've wanted to go to Albarracin for quite some time, but I believe it requires a car. At least I've never uncovered any sign of public transportation. It seems to be a very tiny place. I can't speak to the condition of the roads, but I say, "Go for it". Then come back and tell me what I've been missing!

There are two very interesting places in that general area that I visited in 2016 and highly recommend: Cuenca, which you've already mentioned, and Teruel, which you have not. Cuenca is highly accessible from either Valencia or Madrid because it's on the AVE line, though you then need to take a bus or taxi into town. The AVE tickets are affordable if purchased way in advance, and there's supposedly going to be a ticket sale on February 25.

Teruel was more difficult for me by public transportation. I approached it by train from Zaragoza. There's a highway connection from Valencia, so I guess there's bus service as well. For a regional capital, it is very isolated. As far as I can tell, there is no public-transportation link to Cuenca. If I were willing to rent a car in that area, I would try to hit all of Cuenca-Albarracin-Teruel-Zaragoza.

It may be hot, but it shouldn't be too awful. Zaragoza, in particular, bakes in mid-summer.

If you're interested in modern art, it would be good to have two nights (one full day) in Cuenca, because--aside from its very atmospheric historic upper town--it has two good modern-art galleries. There's also an historic cathedral with an English audio guide. I think there are some English-language walking tours, but I'm not sure about the frequency. I assume there's information online somewhere.

I visited Teruel on a day-trip from Zaragoza only because I couldn't find affordable lodging in the former. There was some sort of big-deal wedding going on, which may have been a significant factor. It's a very interesting town with mudejar towers as well as some modernista architecture. I'd want a night there for sure; I was so rushed that I can't tell you whether you'd benefit from having two nights. It's certainly gloriously unvisited by foreign tourists. In fact, the blog I've linked to says there is a local promotional group called "Teruel Existe". Gotta love it. It would be a real shame to be in this area and not see Teruel.

In Zaragoza I stayed at the business-class Eurostars Hotel at the main train station, Zaragoza Delicias. I don't recommend that. The hotel itself is fine, but there's very, very little in the way of food within convenient walking distance, though there is good bus service from the station to the historic/touristy area where a lot of the restaurants are located. If you're stuck in the hotel at night in the summertime (very low season, apparently, due to the oppressive heat), good luck navigating the short room-service menu. I recommend the cheese plate and the fruit place. You don't want the pizza.

You'll probably want 2 nights in Zaragoza, though I'm just guessing. I hadn't done research in preparation for my stay there, which was really just to facilitate the visit to Teruel, so I certainly didn't see everything I should have. The Aljaferia (located midway between Delicias Station and the historic area) is very interesting and will take some time to see; there's an English audio guide.

With 12 nights ...
• Barcelona - 3 or 4 nights
• Train to Valencia - 3 nights
• Train to Teruel - 3 nights (rent a car for local sightseeing, or book an organised tour to Albarracin)
• Train to Zaragoza - 2 or 3 nights
• ALSA Bus (via San Sebastián) to Bayonne

If you get a car, as well as Albarracin, you might like to include Morella, Daroca, Calatayud.
Cuenca is quite the additional distance I think.

From ... http://www.spainthisway.com/regions/aragon.php

Interesting Towns and Villages of Aragon
Albarracin, a beautiful town which won an international award for historical preservation.
Alcaniz, the castle, now a parador, rises above the town.
Aguero, sited at the foot of the Los Mollos mountains which are more of a big rounded cliff than traditional mountain. Home of two Romanesque churches with excellent carvings.
Ainsa, an attractive medieval town with an arcaded main square. It is considered to be one of the most beautiful towns of Spain.
Alquezar, situated above the Rio Vero canyon the combination of great views and moorish architecture makes this village well worth a visit.
Anso, a village enjoying a revival which sits in the beautiful valley of Anso (with Echo one of the most beautiful in Argon).
Benasque, a village with a beautiful old centre and magnificent views of the Pyrenees. A popular skiing centre and close to Cerler a popular ski resort. Also very popular with hikers.
Cantavieja, this village sits high up on the edge of an escarpment, its Plaza Major is lovely.
Castillo de Loarre, a magnificent fortress built against the side of a mountain and with fabulous views.
Calatayud, a Moorish town with impressive Mudejar towers.
Daroca, medieval battlements and towers make this a very interesting town to visit.
Echo, a beautiful whitewashed village in the valley of Echo, one of the loveliest in Aragon.
Fraga, a medieval town in the Aragonese Pyrenees.
Fuendetodos, birthplace of Goya. It is possible to visit the cottage he lived in.
Graus, town houses covered in frescoes make the town's Plaza de Espana a real treat.
Huesca, the regions capital until 1118 is now a pleasant town.
Jaca, one of the bigger towns in the area and has a historic past figuring Romans and Moors. Its main sight are its sixteenth century citadel and its cathedral.
Mora de Rubielos, with a very well preserved castle and an attractive fortified church, this is an attractive medieval village.
Morella, built on a steep rock with a castle built at the top.
Rubielos de Mora, another very attractive village.
Roda de Isabena, an attractive hill village with a splendid cathedral.
Sos del Rey Catolico, birthplace of Fernando of Aragon who united Spain. A pleasant town in a beautiful setting.
Teruel, Teruel cathedral is a treat and the town boasts five Mudejar towers. A must for anyone interested in Mudejar architecture.
Tarazona, lots of Moorish architecture including a cathedral which is a mix of gothic and mudejar style architecture.
Valderrobres, an attractive town with a lovely old quarter.
Zaragoza, called Saragossa in English, is the capital of Aragon, fifth biggest town in Spain and an important centre of beautiful Mudejar buildings.

Cuenca and Teruel are only 90 miles apart by road, but maps suggest that the road is one of those twisty-turny things. As evidence that Google Maps is not to be trusted for driving times, it indicates that the drive can be made in just under 2 hours, whereas the more realistic ViaMichelin estimates 2 hr. 39 min.

Thanks for all your replies and encouragement. David, that is certainly quite a list of towns you provided, do you actually know all those towns? If so, which are the most beautiful and interesting, and are any on the general Valencia to Zaragoza to San Sebastian route?
I have, in the past couple of days, committed to this trip, Turkey and Spain, and was able to schedule all my flights via minimal United mileage. Not bad, although my schedule needed to be a little flexible to adjust to the flights available to me.
In Spain, I will arrive June 9 and spend the first three nights in Barcelona (boy, hotel prices and availability are tough this year, in 2016 I spent a week in a great small B & B for 70 euros per night, but it is no longer a B & B; now a big apartment; I guess Liz got tired of being up to make breakfast everyday, I would too.)
So, I will then have 10 more nights before my Basque Tour begins, to cover generally Valencia, Cuenca. Teruel, Abarracin, Zaragoza to San Sebastian and up into France. Then I will have 5 more nights after the tour, before flying home from Barcelona (so I may leave Zaragoza for that period).
I have more questions for anyone who knows this area, but I will post more specific pleas for guidance soon.
Take care and be well, and travel a lot, Larry

Sorry it was remiss if me not to credit the source of that list. http://www.spainthisway.com/regions/aragon.php

I can personally recommend Morella, Teruel, Daroca, as well as Zaragoza. Albarracin and Calatayud are places I regret missing due to time constraints.

Check out Pensio2000 as an alternative. I've stayed there on both my visits and would go back again.