I`ve been to Spain many times as a kid, but memories of those trips aren't really fresh enough to be put on this website...
Based on a one week trip to the Barcelona area in September 2005.
Iberia often has surprisingly cheap flights on offer, even at short notice. From the airport, there is very good train connection to Barcelona (about 25 min). Generally speaking, the whole transport system in and around Barcelona is very fast and efficient, and also really cheap. Once in Barcelona, get a 10-trip ticket for the metro/buses (which is a much better deal than the various several day passes).
Probably the cheapest place to stay is Hostal Malda (Calle del Pi 5, Tel. ++34 93 3173002, single: 15 Euro in 9/2005). However, they don't accept reservations and it's kind of hard to get in. I would recommend making a reservation for your first night at another place and then see if you can get into Hostal Malda the next morning. Also, some rooms at Hostal Malda are ok while others clearly aren't, so if you don't like what's available, you can always stay at your original place. A very good though somewhat pricey option is Hostal Lausanne (Avenida Portal de L'Angel 24, Tel. ++34 93 3021139, single: 30 Euro in 9/2005). Ideal location in a pedestrian shopping street (busy during the day, quiet at night) right off Placa Catalunya, i.e. close to the metro, the tourist office and the El Corte Ingles department store (with a large supermarket on the lowest floor and a cafeteria on the highest floor).
Even though I'm generally not a big fan of take away chain restaurants, I have to mention two really good chains: Pans y Company for sandwiches and another one for falafels (with a funny name that I don't recall) where you can create your own falafel at a buffet. Despite being a chain, these vegetarian falafel places are also a meeting place for people with an "alternative" mindset...
Old town
Barcelona is a really great city, but air pollution and noise can be a problem. Try to stay away from the big avenidas and their crazy traffic and focus on neighborhoods with pedestrian zones, small lanes and squares like Barri Gotic and La Ribera. Las Ramblas, a long tree-lined avenue with a pedestrian part in the middle and heavy traffic on both sides, is very famous, though I thought it was a tourist trap (except for the nice Boqueria market and for the part north of Placa Catalunya which is quieter and less touristy). Placa Reial is very beautiful, and there are many other nice small squares in the old town such as Placa del Pi, Placa Sant Felip Neri, Placa Sants Just i Pastor, Placa Santa Maria. The patio in Museu Frederic Mares near the cathedral is a great place for a siesta break. Try to visit the cathedral and its very nice rosario on Sunday morning during mass to get around the entrance fee. Around 12 p.m. on Sundays, you will also get a chance to watch the Catalans' national danse and music (sardana) on the main cathedral square. There is supposed to be a good view of the old town from Torre del Rei (closed for renovation in 9/2005). To walk around the old town, get away from the most touristy parts near the cathedral and venture into the more southern part of the Barri Gotic (south of Placa St. Jaume) and the area around Santa Maria del Mar in la Ribera.Modernism
One of the best things about Barcelona (particularly for Art Noveau fans like myself) are the various buildings in the Catalan style of Art Noveau (modernismo). Gaudi is the most famous artist of that movement and also something of a Catalan hero. If you are into modernism, check with the tourist office for a special pass that allows you to visit many of the buildings for a reduced fare. Absolute musts are Casa Battlo (both the inside and the roof) and Sagrada Familia (don't even think about taking the elevator, walking up the stairs to those towers is the best things about the whole visit). To get an idea of what Sagrada Familia will look like if they ever finish it according to Gaudi's plan, take a look at the very good paper model in the tourist shop across the street from the southern front (the main construction site). From Sagrada Familia, it's only a short walk to the very interesting and beautiful Hospital de la Santa Creu i Sant Pau which was built by another artist of the modernism movement, Domenech i Montaner. Another famous modernism sight is Parc Guell which can be reached by taking the 24 bus from Placa Catalunya (this will save you the long walk from the nearest metro station through an ugly and noisy neighborhood). Parc Guell features some more Gaudi stuff and, most of all, an amazing view of the city (particularly from the hill of the three crosses). If you still haven't gotten your fill of modernism, you can visit the very nice Palau de la Musica, possibly Palau Guell (closed for renovation in 9/2005) and maybe Casa Mila/la Pedrera (though to be honest, I thought it was pretty ugly from the outside).
Gracia
By far my favourite part of the city, away from the tourist crowds. This is where I'd live if I was living in Barcelona. Gracia used to be a village until it was "eaten up" by ever-growing Barcelona and it has retained that village feel. There are many small tree-lined lanes, cute little squares, each with its own character, a great ice cream place (on Placa Revolucio Setembre 1868) and an artsy movie theatre (in Calle Verdi). Gracia is a meeting point for artsy, alternative, free-spirited people and a popular place for students to go out. The area has a strong community spirit, so put away your tourist map and camera and just blend in (after walking around Gracia for less than an hour, I already got invited to participate in an art project which I thought was hilarious...). For orientation purposes, the nice part of Gracia is basically the triangle between the metro stations Diagonal, Fontana and Joanic.
Activities
Get the Guia del Ocio, Barcelona's weekly agenda for movies, theatre, music, restaurants etc. (in Spanish, available at most newspaper agents).
Barcelona's beaches are very urban and the water is not the cleanest, so head out of the city to go to the beach. Platjes de Castelldefels, only a 25 minute train ride away, has a long stretch of beach. A little bit further on the same train line is the cute little seaside town of Sitges, with very nice buidings and a great promenade (plus some rather small and short beaches). During the week in the off-season, Sitges is beautiful, clean and calm. The peace and quiet and the clean air make it a nice escape from the city and it's definitely worth staying there for a couple of days (the long beach in Castelldefels is only 10 minutes away by train, and Sitges is a great place to walk around in the evening). Hostal Parellades is probably the cheapest place in town and very nice (Calle Parellades 11, ++43 93 8940801, single: 28 Euro). Beware that Sitges can turn into a rowdy party place in the main season and on weekends, so Hostal Parellades might get too noisy then, as most of its rooms are facing one of the busiest pedestrian streets. Izarra (Calle Mayor 24) is supposed to be a good place to eat out in Sitges.