Lisbon is relatively small and you can get around easily by public transportation. A daily pass costs around € 6 and gives you 24 hours on the metro / bus / tram / train. Definitely worth it, especially if you're planning to take the yellow trams, which are terribly expensive.
Taxis are reasonably inexpensive and in general trustworthy. The ones at the airport are known to scam tourists. Make sure that the meter is running. A trip from the airport to the center shouldn't set you back much more than 12 euros.
Lisbon is not a city with major monuments like Paris or Rome. It's mostly about experiencing the city and walking around. That said, the following areas are nice to visit:
- center, Chiado, castle, bairro alto.
- Belém, 20 minutes from the center by tram. Nice monastery,...
- Expo, the old Expo '98 area is full of modern architecture. Not everybody's favorite, but
- Sintra - [description] 40 minutes by car
- Óbidos, Caldas da Rainha, beach - one hour North of Lisbon lies Óbidos, an old medieval town. You can combine this with Caldas da Rainha or a beach. If you want to make it a 2-day trip, you can add Tomar to the list.