Handicapped in Portugal

Choose a hotel in flat part of the city, Sofitel or Avenida Palace.

With only 6 days, you are spreading yourselves far too thinly trying to cover 3 places.

Lagos is hilly, with cobbled streets, so would be difficult with a cane. The Algarve is all about the beaches and golf and November isn’t the best time for either. The boat trips to the caves would probably not be running at this time of year.

Although Lisbon is hilly, there is a transport system there so you can get around.

I would spend the entire time in Lisbon apart from one night in Evora and have a day trip to Cascais or Sintra.

In only six nights, I would definitely skip the Algarve. There are lots of beaches closer to Lisbon and going to the Algarve would take most of two days out of your six, just to travel to and from.
The Baixa area of Lisbon is flat so it’s a good area for your accommodation. And, as stated earlier, public transport is very good and taxis are inexpensive. To see the hilly areas, there are (way too many, IMO), tuk -tuks around town.
If traveling by metro, be sure to check about accessibility. A lot of the metro stops have elevators, but not all of them.

The old joke goes, "if you get tired of walking around San Francisco, you can always lean against it." Well, this applies equally, if not more so, to Lisbon.

In Lisbon, the Baixa is flat, the area around the aquarium is flat, and Belem (outside the center, but with major sights) is flat. But the Alfama and the Bairro Alto are VERY hilly. Plan to take taxis or trams or buses not only to the Alfama and Bairro Alto, but also within them.

Thank you for the suggestions. Does anyone have experience with the Lisbon area boat tours, what company is the best?