While much of Barcelona can be done on foot, you'll soon find that public transport not only gets you from A to B, it also lets your poor feet recuperate a bit.
The best way to get around is by metro. The system is modern and trains are frequent. The downside is that the metro finishes early - at midnight during weekdays and 2 a.m. at the weekend and holidays.
The other options are the bus - there is an extensive bus network in Barcelona and it covers areas that are badly linked by the metro. There are also night buses available on certain routes.
Failing that you can always take a taxi. Much cheaper than most citys, taxis in Barcelona are a quick and cost-effective way of getting around and to and from the airport.