The best way to reach Algarve from Lisbon is by Private Transfer.
From Lisbon, you can request pickup at your hotel/accommodation front door, or airport, with your luggage, directly to your destination in Algarve, without the hassle of public transportation and crowds.
Any destination place in the Algarve can be covered, some examples: Lagos, Sagres, Lagoa, Faro, Lagos, Quarteira, Carvoeiro, Almancil, Albufeira, Portimão, Tavira, Vilamoura, Salema, Ferragudo, Porches, and much more.
There are three options to consider:
OPTION #1: Direct Transfer (Private), no stops (by LisbonGuide / Concierge Portugal): https://lisbonguide.org/private-transfers-portugal/
OPTION #2: Private Transfer, by Portugal Magik Tours: https://portugal-magik.com/tour/private-transfer-from-lisbon-to-algarve/
OPTION #3: Car + Driver at your disposal with Portugal Magik: https://portugal-magik.com/destination/algarve/
You can also follow here this app, for instant quotation and booking: https://lisbon.ridebitsapp.com/central/new_reservation
Enjoy Lisbon and Algarve. Make sure to travel in comfort and safety.