Flying into Lisbon but how may days in Lisbon and what other cities are worth spending a couple of days in and easy to get to by bus or train?
How much time total do you have planned? (Oops, I see 10 days). The places I’ve seen other than Lisbon are Sintra, Évora, and Coimbra, all of which I enjoyed a lot. Lisbon deserves at least 4 nights, I think, and more would be better.
I recommend 4 or 5 nights in Lisbon and 1 or 2 nights in Sintra. We visited Sintra as a day trip from Lisbon, but staying overnight would be better. Haven’t been anywhere else yet in Portugal.
We visited Sintra with a tour group. When we left Sintra, we drove by Cascais and made a quick stop, and drove through Estoril. Both places looked very nice and we commented that it would be nice to spend a full day there. Other popular places (but we haven’t been yet) are Porto, Douro Valley, and Evora.
I recommend buying some good guidebooks, such as Michelin Green Guide, Rough Guides, Rick Steves, and/or Lonely Planet.
I did a 10 night trip without a car last June: I spent 4 nights in Lisbon including first 3 nights plus the last night, and included a day trip to Sintra; 1 night in Tomar (highly recommend); 3 nights in Porto including great day trip tour to Duoro valley; 2 nights (1 full day) in Aveiro (very pleasant but not a "must do" in my view). All were easy and cheap train connections.