Traveling in Portugal without a car

I am in the process of planning our trip to Portugal- have not been yet- there is a great trip report here by Priscilla who traveled without a car to most of the destinations you are interested in.
There are 4 installments- I found so much useful info in this trip report- follow the links thru for all installments:
https://community.ricksteves.com/travel-forum/trip-reports/portugal-our-trip-in-progress

I like to use rome2rio.com to research transpo options- great starting point but use it just for reference.

There is a train from Coimbra to Regua but would mean a transfer in Porto so it might make more sense to start in Porto and do the Douro portion from there. There is also a bus from Coimbra to Regua. As far as getting around the area without a car- the steam train that runs along the river doesn't start running til June so that's out but there are boats and local trains. Other than that you'll need advice from a good guide book or others here with experience.

Train from Lisbon to Evora is straightforward. Some do Evora as a day trip
Train from Evora to Coimbra means a change in Lisbon.

Hope that helps get you started.

I'd suggest 2 nights in Sintra. It's a very long day trip from Lisbon and unless you get up very early, you won't be able to miss the crowds at the Pena Palace. I'd suggest you take Uber (or a somewhat more expensive taxi) from LIS to Sintra, then Uber or train to Lisbon for 4N. I don't think it much matters where you start and where you end.

Evora is a long day trip from Lisbon. Going there on the way to Coimbra means adding travel time as well as all the time and hassles of changing locations. Frankly, I'd consider dropping it and adding time to either Lisbon or the north. a 16 day trip is 15 nights. Sintra 2, Lisbon 4, Coimbra 2, Porto 5. That leaves you 2N for the Douro Valley, unless you are taking nights from Porto for that. From Lisbon, you'd have a day trip to Cascais, and 2 full days in Lisbon . . . not really enough. With another 2 days, that's one more for Lisbon and one for a day trip to Batalha/Alcobaca (gorgeous Gothic monasteries) or somewhere else. There are day trips from Porto too. I liked Aveiro a lot. 3N in the Douro Valley would give you 2 full days there. On a 2-week trip, I think 2 "outliers" (Evora, Douro Valley) is one too many.

Before my trip to Portugal last summer, I spent some time looking into some of what you are interested in. I ended up deciding that getting to Evora from Coimbra or Porto or Regua by public transport, though possible, was too much travel time with too little flexibility so I dropped Evora. You have more days than I did so you might work Evora in as a long day trip or overnight trip directly from Lisbon.
I had also initially been interested in spending a couple days in Pinhao and/or Regua to see the Duoro valley without a car and not on an organized day tour. The train ride is said to be very scenic. I read various blogs on how to do it on your own without a car, but eventually I decided to take a small group van tour (day tour) from Porto that went to a couple family run wineries plus boat ride instead and was quite happy with that. There may be folks with better knowledge, but it seemed to me that unless one hired a taxi or private driver for at least a day, you might be limited to the largest and most tourist sounding Quntas closest to town and that you would need to make private reservations with each Quinta and figure out the transportation and timing. I would be happy to share some of the Duoro 2-3 days without care information I collected but didn't use.

I did nine nights in Portugal last year without a car - flew into Lisbon, out of Porto. I day tripped to Sintra and Cascais by train from Lisbon, and I took the train north to Coimbra (detour in Tomar for a few hours) and on to Porto by train. While in Coimbra for two nights (probably one night too long), I day tripped (by bus) to Nazare, a lovely (if tacky-touristy) beach town I wish I had spent more time in, maybe one night to enjoy it.

I had planned to do what you propose in Porto: train out to the Douro Valley but just for the day. But by the time I got there, it was raining, and I didn't feel much interest in doing that in the rain so I just stayed in Porto.

Porto is charming but small - there is honestly a lot more to do/see in Lisbon. If you have enough planned (day trips, etc.) in Porto, then five nights would be fine but I wouldn't expect to want to spend a ton of time in Porto itself.

Now that the holidays are over, and I have returned to my Portugal trip planning, I'd like to thank everyone for their suggestions. I have altered my plans accordingly, and I really appreciate having people who are so willing to share their ideas and experiences.