What is the best spot in Algarve region?

The Algarve is definitely not my favorite part of Portugal, but the Lagos area is my pick. It's big enough to have lots of restaurants, etc. The downtown area is nice to walk around and the beaches and rock formations are spectacular. You can also visit Sagres, Monchique and Silves from there. If you go further east, (Faro), it's way too built up for me.

The Algarve is all about the beaches and golf and October is borderline beach weather. I like the Algarve beaches in the summer, but at that time of year, I would stay further north. Lisbon, Evora, Cascais, Sintra, Obidos etc should keep you occupied for 8-10 days. I would then head to the Silver Coast if you want some sea air. Somewhere such as Peniche or Figueira do Foz would involve less travelling than the Algarve.

If it has to be the Algarve, make it Lagos or Tavira as these retain some character, albeit with lots of modern developments around. You need a car for the Algarve.

It all depends on what you like to do on your trips. We like to hike / walk and experience the environment and culture / local lifestyle and don't try to see all of a country in 3 weeks. There is always next year. We just returned from 3 weeks in Portugal and spent 4 nights making our way down from Lisbon (Sesimbra, Setubal, Vila Nova Milfontes & Carrapateria) followed by 4 nights in Sagres before heading north via Evora to Lisbon. We needed a car for this part of our trip. One day for a walk along the cliffs, we drove to Salema that Rick Steves recommends but I think it would be too small & quiet for a long stay. The Algarve towns were all quiet in November and although we had perfect weather, just about the only people on the beaches or in the water were surfers with wet suits. I wonder how busy / crazy the Algarve towns would be June to September.
Read Rick Steves' book but he doesn't cover the west coast, plus read Lonely Planet and scan the many web based guides.
Enjoy.

We like Quarteira which is between the west coast and Spain so easy driving distance to most sights. We were there in mid September and stayed in an apartment in the old town, a short walk to the beach, restaurants, shops, etc.. After landing in Lisbon from San Francisco, this was our first stop on this trip so we were a bit tired. We have been there a few times before, taking day trips from there, but we still had two days of day trips planned for this visit. We ended up just spending the time walking on the promenade along the beach, stopping for some pastry, enjoying the sights, striking up conversations, etc. and thoroughly enjoyed our stay there. So whether you decide to use this as a rest stop or as a base to sight see along the Algarve coast, Quarteira might serve your purpose.

Salema on the west, followed by Tavira on the east. Research them both to see if either fits your holiday style.