Accommodations

International Living suggests a couple of websites listing apartments for both sale and rental in Portugal: idealista.pt and casa.sapo.pt.

Of course, another obvious site to check would be AirBnB. Here is an example in Albufeira.

In the hot spots of the Algarve region such as Faro, Lagos and Portimão the rent for a one bedroom apartment is around €1000, but there are many in the Algarve for less than €500, some with an ocean view. These are outside the "touristy" areas, typically smaller towns.

Lisbon

For Lisbon, Rick Steves has a recommendations for everything from bare-bones hostels to higher end hotels. I particularly noted Chiado44 Guest House and Feeling Chiado 15. Rick Steves also recommends My Story Hotels, an established chain of hotels with multiple locations around Lisbon and Portugal (Porto and the Azores). In Lisbon try My Story Hotel Rossio, My Story Hotel Tejo and My Story Hotel Ouro.

(Casa do Barão is another recommended bargain place to stay but is closed for renovation until October 2022.)

Algarve

Rick Steves recommends staying away from the central Algarve towns as being too touristy.

He recommends the smaller towns in the western area of the Algarve like Sagres, Salema and Lagos. He suggests staying in Salema and doing side trips to Sagres and Lagos. An alternative is a nice town in the eastern Algarve called Tavira. Tavira is a handy location for side trips to Spain. Transportation to these locations from Lisbon goes through Lagos. See Algarve Bus or Eva-Bus for bus schedules and booking.

Norm and Tina of This Is Our Retirement recommend the following providers of holiday rentals:
Pestana Hotels
Luna Hotels & Resorts
MGM Muthu Hotels
Alfagar Aparthotels

If staying in or near Faro and flying out of Faro, there is a a good inexpensive hotel near the airport also recommended by Norm & Tina of This Is Our Retirement: Hotel 3K Faro Airport.

For inter-city travel by train try Comboios de Portugal or Deutsche Bahn. For European flights Rick Steves recommends Kayak.

My research for short term rentals in Albufeira:

www.idealista.pt - 2 bedroom with parking in Guia, 83m. €1,000/mth

www.idealista.pt - 2 bedroom with parking, 97m. €1,100/mth

Lagos accommodations:

casa.sapo.pt - 2 bedroom apartment-hotel with parking, 110m. €1,200/mth

casa.sapo.pt - 2 bedroom apartment with parking, 110m. €1,199/mth