The best way to reach Algarve from Lisbon is by Private Transfer.
From Lisbon, you can request pickup at your hotel/accommodation front door, with your luggage, directly to your destination in Algarve, without the hassle of public transportation and crowds.
Prices starting at 325EUR
Any destination place in the Algarve can be covered, some examples: Lagos, Sagres, Lagoa, Faro, Albufeira, Tavira, Vilamoura, Salema, Ferragudo, Porches, and much more.
There are three options to consider:
OPTION #1: Direct Transfer (Private), no stops (by LisbonGuide Team): https://lisbonguide.org/private-transfers-portugal/
OPTION #2: Private Transfer, with sightseeing stops (2h), by Sintra Magik Tours: https://lisbonprivatetours.com/transfer-to-algarve-from-lisbon/
OPTION #3: Car + Driver at your disposal with Portugal Magik: https://portugal-magik.com/destination/algarve/
Enjoy Lisbon and Algarve. Make sure to travel in comfort and safety.