Firstly the good news is that according to recent statistics the quality of driving and in Portugal has steadily improved over recent years. I can vouch for this to the extent that driving in Spain these days is more "exciting" than in Portugal.
The road network is constantly being improved.
In the Algarve there is spinal motorway - the A22 Via de Infante - that runs 140km the length of the Algarve from the Spanish border at Ayamonte in the East to near Bensafrim north of Lagos in the West and 30km from Sagres the most South Westerly Point in Europe.
This motorway, which has free access, means that nowhere in the Algarve is more that 120 minutes from anywhere else. With access to the North via the A2 toll motorway at Paderne, regions to the North and even Lisbon are accessible within 3 hours drive.
From Vilamoura, I would recommend the possible following itineraries:
1)
Vilamoura - Lagos by A22 Motorway (Exit at the Lagos Este Off-ramp)
See Lagos
Then Lagos - Sagres by the EN 125 road
Visit the villages and beaches at Salema and Ingrina
See Sagres & Cape St. Vincent
Return Vilamoura
or
On return from Sagres take the left turn at Vila de Bispo for Carrapateira and Aljezur.
Visit Aljezur
Return Aljezur - Bensafrim.
Pick up the Motorway at Bensafrim
Return Vilamoura.
You probably would need two trips to do the above as Lagos itself needs a day.
2)
Vilamoura - Tavira - Castro Marim
A22 motorway
- Alcoutim.
Another nice itinerary but the second part will be very hot in August.
3)
Vilamoura - Silves - Monchique Mountain.
A22 motorway to Lagoa
Turn left for Silves.
Visit Silves Castle
Follow signs for Monchique.
Car hire companies:
See the many posts on this forum.
I usually recommend Luzcar.
Although their prices appear a bit higher than others they are all inclusive.
You can find their details here:
http://www.solpraiaferias.com/carhirealgarve/index.php?option=en
Hope this helps. Others will hopefull add their comments.