There are no public transport systems that link Lisbon directly with Carvoeiro in the Algarve.
Essentially, you have two options:
1) Rail
Train from Lisbon Gare Oriente or Entrecampos stations to Portimão (change at Tunes Junction) and then taxi.
2) Long Distance Coach
Both EVA Transportes and Renex operate long distance coach services from Lisbon to Lagoa.
From Lagoa one can get a taxi to Carvoeiro.
For timetables, go to my blog at:
http://barlaventostravelblog.blogspot.com/
Scroll down on the right until you come to the panel entitled:
Algarve & Portugal - Travel Transport Links
In this panel you can find links to download in pdf format the train timetable from:
# Portuguese Railways - Winter 2007/2008 - Lisbon - Algarve Faro/Lagos Timetable
# RENEX Bus Service - Lisbon Gare Oriente to Algarve timetable
EVA Transportes service from Lisbon Sete Rios Bus Station to Lagoa: (Click on Image to enlarge)
There are buses from both Portimão and Lagoa to Carvoeiro. However, in Portimão the bus terminus is a long way from the station and major bridge works are delaying traffic substantially at present. That is why I recommend taxi.
You can see a full bus timetable on Cubsur's site at:
http://thomasallen.home.sapo.pt/travel/031.htm
