14 nights in Portugal w/o a car itinerary critique.

First thoughts, could you rearrange so that you travel Madrid to Faro? Then you don't have to backtrack.
I know everyone on the forum says you don't want to do Algarve without a car, but my son and his friends did and enjoyed it, however, you really will only have 3 days with 4 nights. I think if it is a priority for you maybe add a night to Lagos.

Lisbon, - you have a lot of daytrips. Tomar is great, but that is too much public transit time for me to do in a day. I'd either stay the night, or drop it as a day trip. I actually like Evora better, and I think it has a direct train, but I'm not positive.

Coimbra you should do on your way to Porto, but you will have a very long day to Porto from Lagos ( unless you are flying?). There is enough to see in Coimbra to merit staying ovenight too.

Hi RafaFan,
Unfortunately, there are no nonstop flights from Madrid to Faro on Apr 26 and no direct buses or trains. Plan on spending your first three days exploring Lisbon’s Alfama, Baixa, Bairro Alto and Chiada neighborhoods before taking day trips to Belém, Sintra and Évora that will take a minimum of seven nights.
I recommend exploring the city you’re sleeping in first and then adding day trips after you’re more oriented with each place you’re staying. You’ll learn more about what’s nearby in case you need to know where a taxi stand, bus and train station(s), laundromat, pharmacy, etc. are located instead of trying to figure it out having no idea which way to turn.
A few years ago, I took a direct bus from Lisbon to Lagos (3h 45m). I was surprised to see so many skyscrapers (hotels) that line the coast but once I arrived in Lagos I didn’t see any of that. Don’t plan on doing much on arrival day and after a night’s sleep in Lagos visit the Bengali caves. After two nights in Lagos take a day trip to Salema by direct bus (30m). I really enjoyed Salema but I was there during the summer. It sounds like you need to allow three nights in Lagos.
From Lagos take a bus or train to Faro and fly nonstop to Porto.

I mapped out the departure times, durations, and prices and it saved me time and money to retrace my steps sadly (long layover). I found a cheap flight with perfect departure time (not a morning person) from Madrid to Lisbon. Going to Lagos from Madrid would have cut into my Lagos time, I'd rather sacrifice time in Porto.

4/27 Lisbon early afternoon arrival
4/28 Lisbon
4/29 Lisbon Belem
4/30 DT Sintra
5/1 DT Obidos-Nazare as part of Spanish speaking tour
5/2 DT Evora
5/3 Tomar? or more Lisbon
5/4 Train for Lagos noon or 2pm arrival
5/5 DT Bengali

5/6 DT Salema
5/7 DT Cape Sagres? or more Lagos
5/8 Train to Porto 6am or 3pm departure, might have to bite bullet and take 6am train..
5/9 Porto
5/10 DT Coimbra
5/11 Return home

You didn't plan a visit up the amazing Douro River Valley. Porto deserves a couple of days as well as up river.

I don’t drink.

I loved the Douro Valley for its beauty and serenity. For me the wine was a pleasant extra.

Take a look at the sights in Sintra before you decide to do it. I suspect it may not be as attractive to you as it is to many others. If you do choose it, an overnight (or even 2) is a much better option. You could go directly to Sintra from the airport (uber is not so expensive and saves a lot of time and hassle over using the metro and train). From there to Lisbon, either uber or train/metro (it's much easier going to Lisbon than going from Lisbon). Or spend a day from Lisbon to Cascais (I think you'll like walking to Boca do Inferno and maybe the beach there) and maybe even going by bus to

The harbor city of Setubal is often overlooked as a daytrip from Lisbon... it's a pretty short train ride, and once you get here, there's hiking in Arrabida National Park, wonderful Forte de Sao Filipe, our charming Old Town, Parque Bonfim, a fabulous wild bird preserve, a dolphin pod in the Sado River estuary, and great restaurants serving the fish that is Setubal's heritage. We've got beautiful beaches too... and people who live here will hate me for spilling the beans about this place...oh, and we have a very vibrant arts scene as well. And a ferry boat over to the Troia Peninsula with its casino, golf course and 11 mile long beach... Something for everyone.