You can hire a car at Lisbon Airport and drop off in Tavira.
Suggest you look at Auto Jardim - see
http://www.carhireyes.com/autojardim/
If you are from across the Pond, I suggest that you specify if you require automatic shift as most cars in Europe are manual shift.
There is public transport available from Lisbon to Tavira.
You would have to take a taxi from the Airport to Gare Oriente Rail/Bus Station from where you could get bus or train.
For timetables, go to my blog at
http://barlaventostravelblog.blogspot.com/
Scroll down on the right hand side panel till you come to:
»» Algarve & Portugal - Travel Transport Links
»» Continue scrolling down in this section till you come
Portuguese Railways - 2008 - Lisbon - Algarve Faro/Lagos Timetable
or
RENEX Bus Service - Lisbon to Algarve timetable
An alternative bus service also exists from Lisbon Sete Rios Bus Station operated by Rede-Expressos.
There is a link on my blog as well but be warned the site is only on Portuguese.
