Gaudi is the main artistic influence in this city, and his touch is everywhere. On the Passage de Gracia, you can visit Casa Batllo and La Pedrera, while walking along tiled pavements of his design. Note the streetlights along this road as well, also his original work.
Tibadabo is excellent, try and catch the first funiculare up there if you can to avoid the crowds, the views are outstanding.
The market (on La Rambla) is amazing at midday, where you can enjoy Tortilla and Sangria while enjoying the range of smells and colours. Don't forget to explore La Rambla at different times of the day. Amazing and entertaining don't do justice! As you walk along, you pass bird and pet stalls, flower stalls with fabulous arrangements, entertainers with the most ingenious and outrageous of costumes. Towards the bottom end are the artists and wonderful cafes. You will no doubt be entertained by singers, accordions, jugglers etc; a Euro or two normally keeps them happy.
Quick Tips:
Cafes don't seem to usually open before 11am, and breakfast doesn't seem to be a big thing in Barcelona. The cafes on La Rambla can be difficult to find breakfast in, but just off of them are some lovely bakeries which sell coffee and pastries.
Most things run by locals that aren't a tourist attraction are shut mid-afternoon for siesta, so don't get caught out. Enjoy the chance to relax in your hotel or on La Rambla, watching the world go by.
Best Way To Get Around:
Taxis are quite cheap and fares will usually not amount to more than 3-5 Euros, especially getting around the city. Go careful as one taxi driver tried to add a 4 Euro surcharge for taking 4 of us in the taxi. This isn't usual though and should be argued.
One other very convenient way to get around for the sightseeing is using the tour buses (especially Barcelona Tours). Buy a ticket for a day, and you can hop on and off any of the 26 stops, which usually are outside the major tourist attractions. Tickets can be purchased for one day or two, and range from approximately 15 euros in price. A recommended way of travelling.