We were in Lisbon in 2014 and stayed in the Alfama area which we really enjoyed. We used public transport throughout our stay & feel we missed nothing we wanted to do & had none of the potential hassles of having a hire car.
Re Sintra-we used rail & bus. Frequent reliable services, plenty to see. We walked from Moorish Castle to Pena Palace
We trained it to Estoril & walked to Cascais -yes regular trains, we found this area catering more to the British tourists via their bars & menus. Loved some of the sand art
We caught the train back to Belem-from memory not all stop at Belem-In Europe I am iclined to "get museumed out" sometimes but the Maritime Museum was an absolute highlight-something like 17,000 items displayed including royal barges & historic aircraft/seaplanes near our exit