We rent a car when we travel in Portugal. The driving is easy and it gives us flexibility. If we want to stay longer at a site we can; when we want to leave, we do. Regarding your itinerary including the places you didn't list, it might be a bit too busy. In the past we checked into flying into Porto and out of Lisbon and it always cost more than flying in to and out of Lisbon, but we did not factor in the cost of the extra gas driving up and back. Our suggestions are that you first check into which makes more sense in terms of dollars, then plan your route and what you want to see and do. If you start in Porto you can stay for 2-3 days in Pinhao a small town on the Douro River. Near Pinhao are many wineries and the prehistoric rock art sight of Foz Coa. You need reservations for Foz Coa as all visits must be with a guide. From there you can go to Porto for 2-3 days. A good place to stay for 3-4 days between Porto and Lisbon is Nazare. It's a great location to day trip to Fatima, Coimbra (don't miss the old university), Obidos, Alcobaca, Batalha, and more; you'll have to decide which. From there to Lisbon for 3-4 days where you can day trip to Sintra by train. Turn in your car when you get to Lisbon as public transportation is good. If you fly into and out of Lisbon then leave Lisbon for the end of your trip.
If you decide to rent a car, pick out the sights you'd like to see and plot them on google maps. That will help you decide how much time to spend in each town including your day trips. Have fun planning.