How long in the Algarve

As you have a month, I would suggest at least 5 nights in Lisbon. You will get different opinions on staying or daytripping to Sintra. Personally with a month I would stay for two nights. It has lots of sights, can be very crowded and you will be ahead of the people coming from Lisbon.
I would also make a point to see Porto and Coimbra. With at least 3 nights in Porto. I personally really liked Evora, but was there in spring, someone else van maybe say if it is as nice in the fall.
For the algarve, with a week I would only give it a few days myself. I liked the area around Lagos the most.

First trip to Portugal late spring 2019, I spent 4 nights in Sintra before heading to Porto, 4 nights and the start of the Camino. We loved staying in Sintra, there is lots to see and no regrets. I will be in Portugal again in October for a quick trip and this time we are spending 4 nights in Lisbon and 4 nights in the Algarve (but haven’t decided where, also open to suggestions).

The Algarve can be ok in early/mid October, but I have also had monsoon like rain at this time. Some of the restaurants in the smaller resorts close in late September or October.

The sea is cold year round but is warmer towards the Spanish border. The Algarve is my default summer beach destination, but for October, I would focus on the area north of Lisbon - Tomar, Alcobaca, Batalha, Obidos, Santarem and Coimbra. A few nights in Porto and October should be good for the Douro as that is grape harvest time. You have enough time to see the Minho or some of eastern Portugal that is less touristy.

However, as you have a month, you could head to the Algarve for a week maximum. I don’t think it’s worth the relatively (for Portugal) long journey there for much less than a week (5 days minimum). I would only factor in the Algarve if you can’t fill your days further north and Madeira.

If you want to see the Algarve, I would head to Tavira which has been less overdeveloped than my second choice of Lagos. Lagos and the western end of the Algarve is much windier than the east and can be very foggy in October. Tavira is more popular with the Portuguese, whereas most northern Europeans head west. From here, you could take a day trip to Spain, drive north to Alcoutim on the river, just across from Spain or explore the hinterlands - Rocha da Pena has excellent walks. Avoid Portimao, Praia da Rocha, Albufeira and Vilamoura if you want something more characterful.

Thank you for the suggestions. We have decided to stay I. Salem’s for 5 days, only because we house-swapped for a week. So we will be cutting it short by 2 days so we can spend more time in Sintra. We do plan on seeing all the areas you talk of, including 4 days on Madeira. I’m already tired, but I. Sure it will be wonderful!

Salem? Do you mean Salema? You definitely need a car there as public transport is thin on the ground. I have stayed there in October and found that some restaurants were closed and I have had torrential rain. If you haven’t already booked, I would suggest Luz, as there are more restaurants.

RS’s recommendation that Salema is a quaint fishing village is a decade out of date.