We will be traveling from Seville and Cordoba to Lisbon by train. Our initial views, by way of the web, of the Algarve region is so invitingly unique. The geological sites, architectural ambiance, beaches, food, music! Faro? Albufeira? Lagos? Other? Which do you recommend for a home base?
The Algarve isn’t unique - it’s similar to the well developed Spanish coast and is all about the beaches and golf. If you want something quaint and steeped in history, then the Algarve isn’t for you - head to the area between Lisbon and Porto - this area seems to match your requirements.
Many of the places in the Algarve are purpose built resorts that barely existed 60 years ago, so there is less interesting architecture than places further north. Most music you will hear is pop music, not traditional Portuguese fado. The food is similar to that found all over Portugal - simple, fresh grilled meat or fish with chips and salad. The wine is good.
The folded rocks to the west of the Algarve and heading up the west coast are interesting.
Lagos or Tavira are the best bases. I wouldn’t stay in Albufeira if you paid me to go there - far too many stag and hen parties making the most of cheap alcohol. Faro is a working city and the old town is worth spending a couple of hours if you are nearby, but it’s not the best place to stay. Luz, Alvor, Ferragudo and Silves are also worth considering. The west is far windier than the east. The sea is very cold even in the height of summer.
When do you plan to go? Some of the smaller resorts close in October. September should be beach weather, but October onwards is usually more mixed. I have had lots of torrential rain the first week in October. The weather this year is better than average.
I presume that you will be hiring a car, as public transport isn’t that great.
Have you checked the train schedules? Unless something has changed that I’m unaware of, there are almost non-existent train connections between Spain and Portugal. There are buses that are quite comfortable but taking the train is all but impossible. There is one train between Madrid and Lisbon but it takes 10-12 hours. A bus or flying is a much better option.
You can fly TAP Airlines from Seville to Lisbon. Much better option than taking the train, if the train even exists.