Lisbon, Lisbon Portela Airport (LIS), & Sintra
Lisbon airport: The airport is located rather close to the city center, and it is quite easy to take bus #22 to Marquês de Pombal for 1.45€. If you have a lot of luggage, you should take the Aerobus for 3.50€ which goes through the downtown area and all the way down to Praça Comercio and Cais do Sodré on the water. The Aerobus ticket also acts a day pass for the rest of the Carris public transportation system in Lisbon on the day that you buy it.
Public transportation: The Carris website is in English if you'd like to look at bus/metro/tram/ferry routes. A single ride is 1.45€ and a day pass is 3.75€ plus 0.50€ for the Viva Viagem card. Once you have the card, you can then just add money (or perhaps another day pass) to it afterwards. Each single ride with a Viva Viagem card is 0.85€. Make sure to validate the card every time you get on the bus/tram/ferry and of course, you will need it to enter and exit the metro. This card is also valid on the elevators in Lisbon, but only for transport. If you want to go up to the viewpoint on the Santa Justa elevator, for example, you will still need to pay another 1.50€.
Downtown: The major sites in downtown Lisbon are all close enough so that you can walk everywhere. You should only need to take public transportation to get to Belém and the Parque das Nações. The best viewpoints of Lisbon are from the Santa Justa elevator and the Castelo de São Jorge. The neighborhood called Alfama, not far from the castle and the Sé cathedral, is very picturesque.
Belém: Famous for its pastéis, you will also find the Jerónimos Monastery (with Vasco de Gama's tomb), the Tower of Belém, and the Padrão dos Descobrimentos (Monument to the Discoveries) in Belém. Take tram 15 from Praça da Figueira to Mosteiro dos Jerónimos. There is also a stop called Belém directly before this stop, but it is further from the major sites. The famous pastry shop where you can buy the pastéis is across the street from the Mosteiro dos Jerónimos tram stop (in the direction of downtown Lisbon).
Parque das Nações: This is the modern part of Lisbon where you will find the Vasco de Gama mall, Oceanarium, Atlantic Pavilion, Vasco de Gama tower and the site of the '98 World Expo. Just take the red metro line to Oriente, and the mall is directly across the street.
Day trips: A round-trip to Sintra is 3.50€ from Rossio train station so just make sure you have that much money on your Viva Viagem card, and validate it normally as you go through the gate. If you don't yet have a Viva card, then buy a round-trip ticket at the counter and you will pay 4€ for the ticket plus card. The station in Sintra has no gates, so make sure to validate your card (the machines are next to the entrance into the small station building) before getting back on the train when returning to Lisbon. The trains run every 20 minutes, and both Rossio and Sintra are at the ends of the train line, so it's impossible to miss your stop. The trip takes about 45 minutes.
In order to get to the Palace of Pena or the Moorish Castle, go left when exiting Sintra station and take the bus for 4.60€ round-trip. The tourism office is also located near the bus stop. It is possible to walk all the way up to the palace, but it takes nearly an hour and a half and it's all uphill through winding roads with no sidewalks. The Parques de Sintra website has information on admission prices. The palace was 9€ and the castle was 5€ (or 12€ for both) in October 2010.
Another day trip is Cascais, located on the Estoril coast, if you're looking for a beach. You can take a train there from Cais do Sodré train station.