Fly to Faro cheap with:

Flight time from UK is between 2 and 2.5 hours.

From Faro it will take around 50 minutes by car via the motorway.
Get the train from Faro to Lagos (need to take a cab to Praia da Luz, cost around 10 Euro with the bags).
A cab from Faro will cost you around 60-65 Euro.

To rent a car you can do it from or

More information available in request.