Sorry for the delay in reply.
There are various options for driving from Lisbon to Lagos and Faro.
The Coastal route. There are various variants via Setubal, Troia Peninsula, Sines, Vila Nova Milfontes, Odeceixe, Aljezur to Lagos.
The bulk of this route is 3 metre single lane each way asphalt. Transit time to Lisbon dependent on traffic 5 to 6 hours although I have done it in 3hrs30mins when pushing it.Speed is restricted to 90km/hr.
The route is scenic but with curves in many places.
The Motorway Option. There is a main Arterial Toll Motorway (A2) running from Lisbon to the Algarve (Paderne) where it joins the Via de Infante (A22) Algarve Spinal Motorway. Lisbon to Lagos by this route can be done in 2hrs:30mins.
Parallel to that route is the old main route from Lisbon to the Algarve. (IC1) The bulk of this route is 3 metre single lane each way asphalt with passing lanes. Speed is restricted to 90km/hr.
There are other longer routes through the Alentejo including via Beja to Alcoutim to Castro Marim and then back along the Algarve coast to Lagos.
Essentially, to recap there are probably about 20+ variants on this route. Recommending a single route would be dependent on your interests.
Finally, remember when hiring a car to specify automatic shift, as most hire cars this side of the pond are manual shift.
