Someone just asked me for advice about what to do and see (and eat!) in Barcelona, a city I love, where I spent most of last summer. Here are some ideas:
Museums - Articket is a good discount, valid at five major art museums, for the price of two single museum visits. It includes the Picasso museum (in the la Ribera neighborhood) -- an interesting view into his youth and early works. Also included are the Center for Contemporary Culture (CCCB) and the Museum of Contemporary Art (MACBA), which have very interesting exhibitions. They are both located at the edge of El Raval, and the nearby bookstore La Central is great, with an entrance on Carrer (Calle) Elisabets.
For history, I highly recommend the museums on the History of Catalunya (on the waterfront) -- one of the best history museums I've seen, and a good way to understand the complexities of Catalan history, starting from its prehistory, through the Muslim and Medieval periods, and a thorough discussion of the Spanish Civil War, whose painful memory is still very much alive. Also the museum of the History of Barcelona (in the Gothic Quarter/Barri Gotic) is great, including well-preserved ruins of Roman walls, wine production and storage, laundry facilities, and other elements that vividly invoke what like was like there some 2,000 years ago. The Refugi 307 (in Poble Sec) is an actual air-raid shelter from the 1930's and says a lot about living conditions in those dark days.
Some great spots that you can find in any guidebook: the Gaudi buildings Sagrada Familia and La Pedrera (on Passeig de Gracia, on the way to the Gracia neighborhood, mentioned below), and his Parc Guell.
The famous Ramblas - just walk through it once to see the bird and flower vendors and the living statues. Otherwise, it's crowded with tourists, pickpockets, and grossly overpriced restaurants.
Much better places to walk around are the neighborhoods. Some of my favorites:
Gracia, for its wonderful, leisurely plazas; Poble Sec and el Raval for the diversity of its immigrant communities (but in el Raval, hold onto your purse and wallet); Barceloneta, for a seafood dinner (on one of the side streets) and a stroll along the Mediterranean. The Parc de la Ciutadella is a pleasant respite from the narrow streets of the adjacent Ribera neighborhood, and one of the city's few large parks.
I don't have much to say about specific places to eat. Despite its many top-notch restaurants, I found that mid-range food, better suited to my budget, was rather predictable. But I love the classic Catalan dishes of escalivada (peppers, eggplant and onions grilled with much olive oil), esqueixada (salted cod), espinacs (spinach cooked with pine nuts and raisins), pa amb tomaquet (bread rubbed with tomato, olive oil and salt), and a delightful cocktail: dry white vermouth with olives.
There is so much music, art, literature, and other culture in Barcelona-- be sure to pick up one of the many publications in Spanish or English (or Catalan--give it a try!) and find some one-of-a-kind experiences while you are there.
The Metro train system is good: quick, clean, and expansive. Be sure to buy a T-10 ticket, which gives you ten rides (also good on buses) for the price of 7. It is generally very safe, but like on the subways of New York, Mexico City, Paris, or most other metropolis, you should watch your valuables. It runs until midnight most nights, 2 a.m. Friday night, and all night Saturday. Taxis are reasonably priced and not hard to find.
There are many great day trips around the region by public transportation (or car), including the Montserrat monastery, the grand symbol of Catalan-Catholic nationalism. Its a bit creepy to me, but some of the art and architecture is stunning, as is its mountainside perch, and there are nice walking/hiking trails. I love the city of Girona -- it's just an hour or so by train, and has much of the charm of old Barcelona, only much more relaxing. Also Salvador Dali's museum-house in Figueres near the Costa Brava -- I've never been, but I've heard unanimously that it's fantastic. And a bit further on is Cadaques, a gorgeous little beach town known for its artists.
Also I recommend learning a few words and phrases in Catalan, to show respect for the region's first language. "Bon dia," "bona tarda," and "adeu" are a good start, even if the rest of the time you speak Spanish (or English). Locals will appreciate it.