Just booked flights for a trip to Portugal next spring. We will be flying into and out of Lisbon as we used FF miles and couldn’t fly out of Porto like I would have liked. We’ll have 22 full days, March 24-April 14. We typically like to do slow travel and stay in one location for 2-4 weeks, but I’m thinking this trip will be different. We don’t want to rent a car, so trains, buses, etc is how we’d get around. Thinking of starting in Lisbon then going north to Porto(hit the cooler northern part later during the trip), then taking the train back to Lisbon for flight. Thinking of 8 days in Lisbon, 8 days in Porto, and maybe 3 nights each in a couple of other towns like Amarante and Coimbra. Planning on skipping the Algarve region. Any thoughts? Should we stay less in Lisbon or Porto and instead of doing day trips from there add other towns to stay in? I’m not familiar with Portugal so any help is appreciated.

Given your preference for longer stays in fewer towns, a week or so in Lisbon makes sense to me -- there's more than enough to do there and you'd have options for some lovely day trips. I adored Porto, but a week or so there would be too much for me. YMMV. I thought Coimbra a gem, easily worth a few days. I found Amarante charming, but an afternoon was enough for me. As an alternative, you might look at Braga or Guimaraes. BTW, I found the Moon Guide to Portugal especially helpful.

So nice that you have 22 days!

I think that 7 nights in Lisbon is just fine, for me it wouldn't be at all too much, as it has an amazing number of sights and exquisite museums to enjoy. To see all the sights of Belém can take most of a day, especially due to this year's crowds. It would also give you time to take the short ferry over to Almada to have a nice lunch at Atira-te ao Rio or the dining spot next door, Punto Final, both with amazing views of Lisbon's pastel skyline.
Also it would give you time to take the commuter train out to the lovely beachside, immaculately kept town of Cascais. It does have lovely parks, a delightful seaside walk along the ocean and the Paula Rego contemporary art museum, plus it's just a lovely place in which to relax.

As to Sintra, you might perhaps take 2 days away from your Lisbon stay or from your overall itinerary to spend two nights in Sintra , as the crowds in the spring will be very heavy.
Several Fodorites, melnq8, maitatom and progol (look for their fine trip reports) did this and at night had the town more to themselves and could hit the monuments first thing in the morning before the day trippers crowded the sites. The day tripper crowds to Sintra can be tremendous and really overwhelming these days.

I've spent 5 nights in Porto very happily but if you do decide on a week, you could day trip to both Guimarães and Braga from there via public transportation, but not on the same day, as it would be too much.
But.... both of these charming small cities could also deserve 3 nights each. From Guimarães you could reach the lovely city of Viana do Castelo, north of there, by public transportation. We've stayed at the historic Pousadas of both Guimarães and Viana do Castelo. For a very nice trip report about Braga, look for the trip report by ms_go, where she and her husband based.

On the way to Porto from Lisbon I would plan on at least 2 nights in Coimbra, the University town, as it's really charming. It's on the same Alfa Pendular train line, so easy.

I agree with kja that Amarante is good (at least it has been for me) for a half day but no more. We've stopped there several times on our way to the Douro.

Of course, you should also look for Maribel's excellent guides to Lisbon and Porto.
And if you're willing to relocate, staying in Sintra does offer MANY advantages over visiting it as a daytrip.

Are you not planning to visit Evora? In my opinion one of the more interesting places to visit in Portugal, plus a foodie town. We did Sintra and Cascais as a day trip from Lisbon but if I were to revisit, I would choose to overnight in Sintra instead as my cousins had done. I would also recommend a visit to Aviero which isn't all that far from Coimbra. We only did Coimbra as a stop to see the University and the historic Cathedrals. It is very hilly and when we were there in mid-June brutally hot so end of March/April would be ideal. We also really enjoyed the monasteries in Batalha and Alboaca ( I know this isn't the correct spelling) Both were great stops and we were practically the only people visiting. Also loved Guimares, very interesting end entirely different architecture, reminded me of Flemish or even similar to Rouen in the historic center.

If you plan to do your entire trip by public transportation, there is both bus and train service to Evora, where you could spend a couple of nights,
but in my experience, to get the most enjoyment out of the quite large Alentejo region, one of my favorite regions in Portugal, one really does need a car to venture to Marvão and Castelo de Vide in the Serra de Mamede, to Monsaraz and the Alqueva dam, to Estremoz, the castle town, to Elvas at the Spanish border (also a castle town) to Vila Viçosa, the marble town, to go on a cork tour, wine touring, to visit the many dolmens scattered about, etc. In the Alentejo, I could easily spend a week (and have done so), but with a car
That said....
There are a few bus excursions from Evora, however.
https://www.visitevora.net/en/alentejo-tours-evora/

So if you just want to spend 2 or 3 nights sans car, it's possible. We've stayed there, as have many other Fodorites, at the lovely Albergaria do Calvario (it books up far in advance with luxury cycling groups as does the hotel across the street, the 5-star M'ar de Ar Aqueducto, that also takes RS groups). Evora also has a Pousada, a former convent, and a new lodging option in the center, the Moura Suites.

To reach Batalha from Lisbon there is public bus service via Rede Expressos from the Lisboa Sete Rios bus station, but it's a two-hour ride.