I would rent the car only to visit the Douro Valley, although some prefer to visit it using the scenic railway "linha do Douro" or even by river boat. To drive there you should rent a car in Porto, which is close.
Porto is not a city where you'd want a car. Go there by train from Lisbon (if you fly to Lisbon). You can take an Alfa Pendular fast train. Trains to Porto depart from the Santa Apolonia station, but also stop at the Oriente station (east of the airport). You can also take an InterCity which is not that much slower (If using IC take the 1st class ticket, as the difference in price is not much and the comfort greater).
Sintra and its palace, can easily be reached by train from Lisbon Rossio station.
Obidos is another nice town. A bus is the best way to get there if you visit as a day trip from Lisbon. The bus can be caught at the Campo Grande bus terminal.
Evora, another popular destination can be reached by train also, from the Entrecampos station.
Whatever you decide to do, avoid having a car while in Lisbon or Porto. Those two large cities are best visited without having a car to worry about. To travel around both cities, you can use Uber, which is very inexpensive, or also buses and trolleys.
If you want to rent a car and visit Portugal by car, rent after you visit Lisbon and return it as soon as you arrive in Porto (or viceversa), assuming those cities are at the beginning and end of your itinerary (which is the way I would organize the trip). You can see Sintra while you stay in Lisbon, since they are so close.