Firstly, it is all eminently possible within the time frame outlined.
Assuming you are arriving Lisbon Airport, picking up your car from there and leaving for the South straight away, I would suggest that you proceed straight to Tavira via the Vasco da Gama Bridge, A2 toll motorway and the A22 Via de Infante motorway.
Viamichelin.com gives the following details:
Costs 44.79 EUR
Toll 17.85 EUR
Petrol 26.94 EUR
Transit Time 02h52 (Allow 3hrs 15mins with stops.)
Distance 299km
As regards the return, I suggest that you use a combination of the A22 and EN 125 Algarve coastal arterial roads to visit the places you wish to see. I recommend visiting Silves and Lagos definitely.
I suggest you might want to base youself in Sagres while exploring the Western Algarve. I can recommend the Albergaria Mareantes as a medium cost quality B&B. To make a reservation, contact Maria de Jesus via the details shown on this page.
http://www.paraisolpraia.com/index.php?content=contact
For the return to Lisbon, I recommend that you follow the N268 from Sagres through Vila de Bispo to the Alfambra junction, then the N120 through Aljezur, Rogil, Odeceixe, Vila Nova Milfontes. Depending how often you stop this should not take more than 4 hours.
North at Vila Nova de Milfontes, turn left at Brunheiras and visit Porto Covo, awind swept beach on this Alentejo coast.
Then head North to Sines. I would not recommend staying in Sines as it is a port town. Head North from Sines up the N261 - 5, the N 261, and the N 253-1 to Troia.
There are hotels there or you can cross the Sado Estuary on the car ferry and visit Setubal, (recommended). There is an Ibis Hotel just out of Setubal in the N10 heading south which I can recommend. There is also a Novotel there as well.
From Setubal you are only 40 minutes by motorway from Lisbon Airport via the Vasco Da Gama Bridge.
I often drive Algarve - Lisbon - Lagos by both routes and a non-stop transit time takes me no longer than 4 hours.
If you want to reduce costs, take the I would suggest that you proceed straight to Tavira via the Vasco da Gama Bridge, A2 and leave the toll road at off ramp 6 - Marateca. Get onto the N10/IC1 south and continue till you come to the A22 Guia intersection. Follow the on ramp for Faro and continue till you come to the Tavira of ramp.
Note that the speed limit on the N10/IC1 is 90 km/hr. The are regular police checks. Allow an extra 75 minutes at least if using this route.
Hope this helps.
