I must say that unless you are familiar with the roads in big European cities, you may want to opt for taking a bus or cab from the airport to the hotel and then using the metro or other public transportation to move within the city. First of all, the signs are in Catalan. I'm fluent in Spanish, and even I had a hard time figuring them out (plus, I’m not the best navigator out there). Second, if you miss a left turn, you have to go all the way up to the next roundabout to turn back. We spent 4 days in Barcelona and after the first day of driving we decided to leave our car parked in the garage and use the metro system.
The metro is inexpensive and easy to use. You will find a station at every major block. Get a multiple-trip pass (10 or more, depending on your needs). We purchased one 10-trip pass and used it between the two of us. Public transportation is the way to go. Also, taxis are relatively inexpensive within city destinations.
If you opt for visiting other cities outside Barcelona, many of the car-rental companies have offices within the city. Rent the car on your last day in Barcelona and save the extra "dinero" from the rental and garage fees (as I learn from my mistakes). When renting, I recommend sticking with American car-rental companies. We used Alamo (partnered with National in Spain). For your own peace of mind, get the extra insurance. I’ve seen the way people drive over there . . . Also, make sure to inspect the car before leaving the lot and report any damages to the attendant.