There is no railway station at Faro airport ( yet ) so you would have to take a taxi or bus into Faro to catch a train to Lagos- the bus and train stations in Faro are within a couple of hundred yards of each other.
The rail service along the Algarve has recently been upgraded and trains to Lagos now run almost hourly from Faro.The fare was in the region of €6.40 earlier this year and the journey takes about an hour and threequarters.For full details visit the unofficial website compiled by 'cubsur' of this forum,at :- http://thomasallen.home.sapo.pt/travel/006.htm
Hope this helps a little.