The Algarve

Lagos is very nice. Large enough for plenty to do. Beaches close to town. I would start there with research.

Faro is larger, but can’t speak to it. Tavira might also be a possibility.

My family live in Alvor, once a fishing village. I’ve spent many months there over the years. It’s a coastal town, and maintains it charm.

Easy walking to the beach from the village. It has everything you need, supermarkets restaurants shops, a local vibe outside high season.

Transport can be an issue in The Algarve. We generally have a car, but taxis are well priced.

We just returned from Portimao. The beaches are beautiful with several access areas to walk stairs down from cliffs to sandy beach spots to use. We stayed along the boardwalk, water side with beautiful view of water and sunsets. Close to bakeries and supermarkets with what groceries and supplies we needed. Lots of restaurants to walk to and nice spots to get a drink or gelato and people watch. Walked to the marina for a guided boat tour of the coast and caves. Not crowded at all in beginning June. We were surprised.

Denise
I second Lagos. I lived there for quite some time and always felt very safe. There is a nice mixture of locals and tourists and plenty to do and see. The beach Is gorgeous, wide and sweeping and lots of walking. The Marina area is pretty (and pleasant for happy hour drinks) depending on your vibe. The bus service is ideal and easy to get to places from the central bus station. Short and long distance.
There is a smaller “town beach” which is handy if you’re in town and want a quick swim.
Be aware it’s quite windy in the western end of the Algarve, tho should be fine in Sept/early October.
There is a local market on Saturday mornings adjacent to the bus station.
A regular service runs to Praia de Luz, a nice town, with a low key atmosphere.

Alvor is pleasant too. Less to do, but gorgeous beaches and a pretty town, tho geared more (in my opinion) to tourists. There is a nice smallish Sunday morning market adjacent to the waterfront. Alvor has a wooden boardwalk which has gorgeous views if you enjoy walking. Alvor can be reached by bus from Lagos. Check the timetable first, Sundays services are fewer and out of season less frequent.
Hope the comments help.
Enjoy your trip.

Just a footnote. Check dates for Portugese national holidays! I’ve been caught out quite a few rimes. They are not always adjacent to the weekends and of course this will affect bus timetables!

Just to mention, it may be getting coler during October, so think about the beach!

So does that mean October is better for the beach or too cold for it?

We were in the Algarve in mid October a couple of years ago. I thought it was too cold to swim but I am from Florida. There were some people swimming.

That said, I thought it was a delightful time to visit. We had a car and enjoyed wandering around. It is perfect weather for hiking which we did.

We went to Lagos and do agree it would be a good base without a car. It was not too touristy in October either.