Public transport isn’t that great in the Algarve and car hire is easy and cheap, particularly for 4 people. I have always hired at the airport, but there will be car hire offices in Portimao, which is the major port on the Algarve.
Three nights means two days. Day 1 - drive a loop from Portimao to Monchique up in the hills, across to Aljezur and explore it’s castle then at Carrapateira, turn off the N268 and drive towards Bordeira beach and park on the right hand side in the car park and take a walk on the stunning beach and look at the amazing rock formations. I would give Sagres and the Point a miss as it’s always bleak and windy, but you may want a quick look, just to say you have been there. Salema - it’s been ruined by over development, but you can loop through it and back onto the N125. Lagos can easily absorb a couple of hours of your time before you return to base.
Day 2 - head to Silves and explore its castle, then head over to Tavira, which has a different character to the western end of the Algarve.