In our trip to Lisbon, we rented a car one day and drove to Sintra and Cascais, which are both pretty near from Lisbon, and are easily affordable in a one-day trip.
Sintra is a town with plenty of touristic attractions, mainly palaces and castles, but also some natural parks. We devoted a couple of hours in visiting the Quinta da Regaleira, which is an estate comprising different buildings including a palace, and a park with impressive fountains, lakes and tunnels, with . For me, it was like Parc Güell in Barcelona, but I liked this one more.
Definitely, a must if you go to Lisbon for more than 2 days.