Central Barcelona is compact enough to be walkable. La Rambla, Old Port and Gothic Quarter are best visited on foot – in fact some of the most picturesque streets and squares in Gothic Quarter are too narrow for vehicles anyway.
Further attractions can be very easily reached by metro, which is clean, air-conditioned and efficient. Single trip ticket costs 1.45 EUR and allows changes of lines. However, if you plan to make several journeys, consider buying a T10 ticket, which is valid for 10 trips and costs 8.25 EUR. Quick calculation shows that with the T10 you save 6,25 EUR for every 10 journeys!
T10 can be used by several people travelling together - simply send the card through the machine at the entrance to the metro as many times as there are people in your group.
Further advantage of T10 is that not only it is valid throughout Zone 1 (where most of the tourist attractions are located), but also for a trip to/from Barcelona airport by RENFE train (it is NOT valid for airport express bus!).
T10 can be purchased from ticket counters at metro stations and from the machines.
If you plan to make journeys outside of Zone 1, you will need to buy a different T10 ticket. It is also possible to buy Barcelona Card which includes unlimited number of journeys within certain period of time and discounts on entrance fees for a number of attractions. It is not cheap though so it's only worth buying if you plan to do some serious museum hopping and/or metro riding.
Further details on Barcelona public transporta and latest ticket prices are available at Transports Metropolitans de Barcelona.