One full day is normally enough to explore all major sights. Take a 10-minute walk from “Palacio Nacional” to “Quinta da Regaleira” – amazing place. Plan about 2 hours to explore some building there what is the most important various caves and tunnels.
Another interesting place to go is “Castelo dos Mouros”.
Where to stay: if there are free rooms stay in Hotel Nova Sintra, you will find there extremely helpful staff, beautiful hotel (feels more like visiting friends) with lovely terrace. Delicious breakfast is included in the price. Free wi-fi available. There is no free parking, but hotel’s staff will explain you where you can park for free (actually the street parallel to the train station, just 3-4 minutes walk to the hotel). And it was really nice to receive a “thank you letter” from the hotel after the stay.
On the way to/from Sintra visit Capo de roca (about 30 minutes driving from Sintra). It is the most western point of the Europe.