The Lisbon Metro system, or Metro Lisboa, will soon include some access to the airport.
Between Lisbon-Portel Airport (LIS) and city center:
Carris Aerobus 1 (bus) WEBSITE
The airport is in an urbanized area close to the city center, so the trip takes less than 30 minutes.
Don't forget to stop by the information kiosk at the Lisbon Airport for free books and maps.
Maps are free at the airport and any tourist kiosk or hotel desk. Any map is fine, but the handiest are the ones from the tourist kiosks. For an official online version of the transport map, click here.
Main transit systems:
Carris (bus and streetcar)
Metro Lisboa (rail)
Perhaps the three most useful rail lines for tourists are the  streetcar line to Alfama, the  tram line to Belem and the Metro [Vermelha] to the Parque das Nacoes.
The majestic arch at the Praca do Comercio, which also serves as a transit hub in central Lisbon.
The main attractions of Lisbon can be grouped into just a few walking sections. Baixa-Chiado: Restauradores > Rossio (Praca Dom Pedro IV) > Santa Justa Elevator > Carmo Archeological Site > A Brasileira (with seated statue of Fernando Pessoa) > Praca Luis Camoes > Pombaline Lower Town > Praca do Comercio. Alfama: Lisbon Cathedral > St George's Castle. Belem: Monument to the Discoveries > Jeronimos Monastery > Pasteis de Belem > Belem Tower. Oriente: Centro Vasco da Gama > Parque das Nacoes > Vasco da Gama Tower > Lisbon Oceanarium > Aerial Cable Cars
Note: The 25 de Abril Bridge is well viewed from several places, including Belem and Praca do Comercio, and via a well-angled stop along the  tram route. The Cristo-Rei monument across the river in the city of Almada is also viewable along this route, although quite distantly. The Vasco da Gama Bridge, longest in Europe, is viewable from the Parque das Nacoes complex. The Aguas Livres Aqueduct is close enough to the Campolide rail station for a good view.
Tour and commuter ferries depart from the Cais do Sodre station.
Further afield: Edward IV Park, Campo Pequeno Bullring, Aquas Livres Aqueduct, Cristo-Rei Monument, the Gulbenkian Museum and the beach resorts of Estoril and Cascais via the CP train.
Line 28 Streetcar:
Praca Luis Camoes <> A Brasileira <> Praca do Comercio <> Lisbon Cathedral <> St George's Castle
Line 15 Tram:
Praca do Comercio <> Pasteis de Belem / Jeronimos Monastery <> Belem Tower
Metro Linha Vermelha:
Aeroporto (end) <> Parque das Nacoes (use Oriente)
LOOK HARD: Watch for this nostalgic urinal along the walls of St George's Castle in Alfama.
And yes, it's possible to use Lisbon as a base for making day trips to other Portuguese cities, such as Fatima and Oporto, using the CP rail network.