Mid-to late June 10 day Portugal

Would add Sintra at least 2 nights: Pena Palace, Moor Castle, National Palace, Quinta da Regaleira, town of Sintra. The train tickets can be reserved 2-3 months in advance for promotion fares. At least 3 tickets bought together you get an additional discount. I would start in Lisbon then head up North, either Porto then Coimbra and finish at Sintra. You could spend your final night in Lisbon close to the metro red line that takes you directly to the airport. FYI- metro doesn’t start running until 6:30, may not work for early morning flights. Never went to Belmonte; try Rome2rio for transportation help.

I traveled solo last May to Lisbon, Coimbra, and Porto, using trains. I stopped for a few hours in the town of Tomar on the way north to Coimbra. I did a quick day trip from Coimbra by bus to the beach town of Nazare - loved it, wish I'd spent at least a night there! Two nights in Coimbra was one night too many; it's a nice town just really small; the university at the top of the hill is the highlight but can be seen quickly with a tour. (There are Roman ruins somewhere outside of Coimbra but I didn't bother with them.) Doing it again, I would have simply checked out of the hotel in Coimbra after the first night and overnighted in Nazare, then taken buses or trains north to Porto.

Porto is not very big (but much bigger than Coimbra). It's lovely in parts but there isn't a lot to do in terms of museums etc. especially on the rainy days I spent there. Lisbon by contrast has much more to do indoors on a rainy day. Maybe you'll get lucky in June and have perfect weather, but you never know. I had planned a day trip out to the Douro Valley but because it was rainy, I didn't bother.

Both Lisbon (especially) and Porto are pretty touristy now. Public trams in Lisbon were packed to the gills with tourists. It seems the city has experienced a tourism boom and public transportation still hasn't caught up, so be prepared for that. Some people use Uber. (I didn't.) Otherwise, public transportation is good to get you anywhere e.g. out to Sintra or Belem or Cascais by train.

Can't help you with multi-person discounts with the trains, as I traveled solo. You can buy train tickets in advance on www.cp.pt/en/ (they also have a smart phone app) a few months out, but they aren't super expensive at the last minute; I guess with six people the savings could add up, but I bought mine at the last minute. They aren't as expensive as say French or Italian high speed trains last minute, so it didn't matter much to me.

More info on the trains in Portugal:

https://www.seat61.com/international-trains/trains-from-Lisbon.htm

I'd try to fly open jaw ("multi-city") into Lisbon out of Porto or vice versa, which is how I did it (through Madrid both ways). Not sure which direction is best north to south or south to north or whether it matters. I'd probably base that on the flight connections.

Try to visit
Sintra
Evora
Obidos
Navarre
Take the train up the Douro River Valley or take a cruise up the river.

Thanks for all the great information. I will check out these suggestions.