We plan to go for 2 weeks in mid June 2026. I have been told Azores and Algarve are must sees but I would love some tips from others who are familiar with Portugal. We like to bike, mild hikes, adventure (ATV/excursions etc). We like to typically chill a day, then sightsee/do excursion, then chill/travel and repeat. We will most likely rent a car. We are up for air bnb's or hotels. Prefer local flare to tourist traps, but we are tourists after all and do like to see the "must sees." Don't love group tours so much, usually prefer to go see stuff on our own or in small groups. We are very social though and like meeting others, just not in such a formal organized setting. We love wine and prefer nature to architecture. Hope that is a good starting point to gather some feedback. Thanks!
You mention local flare. In his guidebook, Rick Steves features what he calls the Alfama Stroll and Baixa Stroll, both in Lisbon. The Alfama Stroll is 2 hours long along tangled, medieval streets and takes you to eight sites with detailed explanations of what you're seeing. Also about two to three hours, the Baixa Stroll in downtown Lisbon takes you to 13 sites, also with detailed explanations of what you're seeing.
A prominent site, the Jeronimos Monastery in the Belem district of Lisbon is wonderful. I've been to 24 countries and have never seen a more interesting monastery than Jeronimos. Take the public bus from downtown Lisbon and you'll go right under the 25 de Abril Bridge. For me, it's even more awe inspiring that the Golden Gate Bridge.
I loved the Livraria Bertrand in Lisbon, what is reportedly the world's oldest, continuously operating book store, dating from 1732 and thus older than the United States. It was fabulous! There's a bohemian coffee shop across the street.
The only other town I saw in Portugal is Faro in southern part of Portugal. I switched buses there between Lisbon and Seville, Spain. Faro is only a resort town that attracts locals and British, but I loved it.
First, I would ponder whether you want to include the Azores or not--to me, it is a different trip, and there would be nothing wrong with spending two weeks there. Scope out the flights, and determine what you would have to give up in Portugal to include it (it's the travel time trade-off). You could easily plan a more conventional sightseeing trip throughout Portugal that intersperses nature stuff, like Algarve.
RS book may not include the Azores--look for a Lonely Planet, Rough Guide, or DK guide.
We primarily visited Lisbon and Porto so this may not be our interest to you but here is my Trip Report from last year that may have some useful nuggets. https://community.ricksteves.com/travel-forum/trip-reports/trip-report-part-1-hills-and-castles-portugal
Unless you are able to get a flight from Boston to the Azores and then on to Portugal, I would agree with the prior poster that the Azores should be a separate trip. If you are interested there is a post at the top of the Portugal section on the azores.
June can be busy in Portugal, and fairly hot. If you can, see if you can fly into the Algarve, and out of Porto. Then I would personally prioritize the Algarve, Lisbon, Douro and Porto, with your interests. You would probably want a rental car for the Algarve, but could drop it in Lisbon, take the train to Porto and then decide how to tour the Douro ( for the wine of course).
I think with 2 weeks you could add on a stop in the Azores in addition to visiting the mainland. I have done it with less time then what you are going for.
I went two years ago and stayed on Sao Miguel. I did not find the area really touristy at all. A local guide told us that it had mostly been people from the US who had family there that would visit and now more recently it's growing in tourism. Sounds like with the outdoorsy stuff you like Azores would be a good option. Madeira would also be a consideration. The hiking and levada walks were amazing.