The easiest way to travel is a direct flight to Faro. There are several inexpensive flights available, please consult the site www.ana.pt (Faro airport).
Alternatively, you may fly to Lisbon, and then take the train or bus to Faro. For the flights to Lisbon please consult the site www.ana.pt (Lisbon airport).
From Lisbon you may proceed to Faro by train. It costs around 20 EUR, for more information please see the page of Portuguese Railways. From Lisbon airport you can reach the train station Oriente in few minutes by taxi, it should cost around 10 EUR.