Barcelona has a very complete system of public transportation that makes it very easy to get around the city.
Currently a single ticket for metro or bus costs 1,05 €, but you will probably prefer to buy one of the travel cards.
A T10 (ten journeys) costs 5,80 €. Each journey gives you the right to use any combination of transport inside the city (metro, bus, and city train) for the duration of an hour and 15 minutes.
This system is based on different zones; if your hotel is outside Barcelona, you can acquire a T10 for 2 zones (11,70 €), 3 zones (16,10 €), etc.
Be careful, since the Renfe train system is not completely integrated in Barcelona for this system. The other local train system, Ferrocarrils de la Generalitat de Catalunya, has total integration in this system.
Instead of the T10, you can also buy a T-Day, which allows an unlimited number of journeys in any means of transport during 1 day (until service closedown) and costs 4,40 €. A 3-day pass costs 11,30 € (not valid for Renfe).
The Metro will leave you close to almost any place you want to go to in Barcelona. Certain zones, though, are easier to reach by bus.
You can see online maps for the metro and train systems following these links:
You can ask for a map of the bus system in the TMB information points at the stations of Universitat, Diagonal, Sagrada Família or Sant Estació, but they're usually available at every station ticket office. You can also ask for it at the tourist information offices.