It really depends what you want to do and at what time of year you want to come! Portugal is a lovely and diverse coutry with lot to see.
Faro is situated in the Algarve region in the extreme south of the country. It is a today primarily one of Europe's main holiday destinations. If sea and sun and sand is what you want then start here and then take a long distance coach or train up to Lisbon and perhaps further North.
Places in the Algarve to consider visiting which retain their Portuguese character include Tavira, Lagos, Silves (inland), Salema and Sagres.
Lagos, (where I live :) ) is an historic town from where +500 years ago some fo the Portuguese voyages of discovery set sail to open the route to the Far East via the Cape of Good Hope. It still retains its old city walls and charm as well as offering great beaches, restaurants and bars.
You could spend ten days in and around Lisbon itself. An ancient city dating back to Roman times, it is spectacularly located on the Tejo Estuary.
Places to visit in the immediate vicinity include:
Castelo São Jorge
Belem
Monesterio São Jeronimo
Queluz Palace
Sintra & its Palaces
Take the Linha da Cascais train line along the Tejo and visit the coastal suburbs of Estoril and Cascais and their beaches.
North of Lisbon you have so many options:
Coimbra - University City and one of the oldest Universities in Europe.
Porto - Portugal's second city with its beautiful river location on the Douro River and its port wine trade.
Braga
Guimares
Bragança in the Far North.
So if you can give us some more specific ideas about what you like we can be more specific.
The best way of travelling by public transport is by train followed by long distance coach. Not that local bus services in rural areas can be sparse and may only run a few hours a day.
See the following links that may be useful:
http://www.solpraiaferias.com/index.php?content=cheg_en
http://algarvebus.info/
http://barlaventostravelblog.blogspot.com/2007/05/rail-and-river-tours-on-douro-river.html
You might also like to browse the following where there are previous posts on Portugal.
http://barlaventostravelblog.blogspot.com
