Barcelona's airport, called el Prat is about 10 km to the city center. Once there, you can go to the center by train (every 30 minutes, lets you in Sants and Plaça Catalunya), by the bus Aerobus, that lets you in the same places and with the same fequencies, but more expensive and with the possibility of finding traffic jams. Both services finish at 23:00.
There is a quite good taxi service, the only way to reach the center if you come by night. The fares are about 12 euros if you go to the center, but this can change depending of the time of the day and, of course, the part of the city you want to reach.
Of course, you can rent cars there and there is a big parking place, though quite expensive if you let the car for more than a couple of hours.
The el Prat airport is served by EasyJet, and Virgin Air, a part of many non low fares companies.
RyanAir flies to Girona, north of Barcelona. Travel by Bus to the city takes about 90 minutes and costs about 15 Euros return fare.
The main train station in Barcelona is called Estació de Sants, but the most centric ones are Catalunya (only regional trains) and Passeig de Gràcia (some long-distance lines stop there).
Stroll on La Rambla, a tile-covered tree-lined pedestrian walkway.
Stroll along miles of beachfront boardwalk, or sun yourself on the beach.
Wander the Barri Gotic, the largely intact medieval center of the city.
Gaudi architecture, including the psychedelic Parque Güell and the still unfinished Sagrada Familia.
Olympic stadium and village
The Montserrat monastery (about an hour outside of town, but worth the trip)
MNAC in the Palau Nacional has the single best collection of Romanesque art in the world, and a fine Gothic collection as well.
A ticket to The Museum of the City of Barcelona includes access to underground Roman ruins and a complex of historic buildings in the center of the Barri Gotic, as well as being a reasonably good historical museum.
Barcelona has some of the best restaurants and cafes in Europe. For starters, you might try any of the large cafes that line the Passeig de Gracia and the Rambla de Catalunya, just north of the Plaça de Catalunya: nearly all offer a variety of excellent tapas (appetizers).
You can get food from any part of the world in Barcelona, but make sure you try some specifically Catalan food. The great Catalan staple is pa amb tomaquet: toasted bread, covered in olive oil, then smeared with garlic and tomato. The selection of seafood is consistently great, although not a lot of it is local (this part of the Mediterranean is prety well fished out).
El Glop (four locations) offers excellent Catalan meals at a price within most budgets. Allow about 20€ per person, although you could get out of there for half of that if you let the price dictate your choice of dishes.
Pollo Rico. Calle Sant Pau 31. A fantastic and cheap grilled chicken place buried in the sidestreets east of La Rambla.
Barcelona offers a remarkable arrangement of accomodations, from cheap, decent "hostal" rooms with the bathroom down the hall to five-star hotels. Here are some that are notable in their price range:
Hotel Gran Via, Gran Via Corts Catalanes 642, Tel. (+34) 933 181 900.
Very central, and originally a palace, the public spaces of the hotel live up to that history. The rooms are large, but otherwise relatively modest. About 80€ per night for a double room with bath
Pensión Norma, up 3 flights of stairs at C. Gran de Gràcia, 87, Tel. (+34) 237 44 78. Modest rooms, some with bath, in the relative quiet of Gràcia, about a kilometer north of the center of town. About 25€ per night.