That list could be very long ... it depends a little bit on how much time you intend to spend there and what you are interested in.
My wife is from Barcelona and we just bought an apartment there, I think I might be able to help you a little bit.
Couple of quick ones top of my head without knowing your preferences:
- Air B'nB is huge in Barcelona. Not beloved by the people living there (they say it drives prices up and people are too noisy), but you can find great options and IMHO it's nicer than hotels. If you go for hotels, google "Barcelona hotel rooftop or rooftop pool" if you are into that stuff and willing to shell out some money. There are a couple of great places you can find that way.
- Unless you are into the uber-touristy stuff, avoid LaRambla and the areas close to it. It's tourist trapalicious. The Bari Gothic is a must, and the area on the other side of la Rambla quite interesting as well (used to be the red light district). I would not miss La Bouqueria, the large market off la Rambla (or really any other food market), but that's about it. Maybe Liceu if you are into opera and classical music.
- Go up the Montjuic, either walk from Plaza Espanya or take the Funicular. Great views. The other mountain (tibidabo) you can avoid, unless you like small amusement parks.
- There is no such thing as typical Tapas in Barcelona. Tapas are not from there. If you want to try typical Catalan dishes, have Escudella (soup), my favorite the Esqueixada (salted cod salad, incredibly good in warmer months), Escalivada (toasted bread with cooked but cold eggplant, garlic and olive oil, rasted peppers and anchovies on top - it's so simple yet so amazing), catalan spinach with pine nuts and raisins, butifarra negra (blood sausage, either with rice inside, with onions, with both or with sugar), Llonganissa and Fuet (sausages, cold cut), and Fideua.
- I won't be of much help with restaurants, I eat at the best one out there: My mom in laws. But if you just steer clear of the touristy areas, you can get great deals and authentic food at lunch, always go for the menu del dia. Usually that runs you less than 20 bucks even in better places, but you can get some for 8 bucks or so in simple ones. It's almost always three courses and often includes beverage, coffee, or both.
- Depending on your preferences, I can ask my family and friends in Barcelona for bar and restaurant recommendations. I heard good things about Chicoa, Cal Boter and Can Cargol (all for Catalan food)
- Try to see a game of futbol at Camp Nou... FC Barcelona is playing the most beautiful style of soccer you'll ever see (unless you don't care about sports, then don't go)
- Don't miss the Gaudi buildings or Parc Guell, it's truly amazing stuff. I would go to Gracia as well, really nice neighborhood, more residential. IMHO, Eixample, Bari Gothic, Gracia and el Born are the areas you have to walk and experience. You don't have to go to the Olimpic harbor, for example.
- If you have time for a day trip, go to Cadaques and visit the Dali museum nearby, or go to Montserrat ... or to the beach, but you don't have to leave Barcelona for that. Might just be nicer further south than in Barcelona itself.
- Never, EVER tell a Catalan that he/she is Spanish. They will be offended, seriously. They do not consider themselves Spanish and most of them want to be independent. Don't tell them that Catalan is just a Spanish dialect, either. It is not. It's a different language, just with the same origin, same as Italian, French, Portuguese. And don't ask where you can see a bull fight. It's illegal in Catalunya and they see it as a symbol of Spain, so it's frowned upon and even hated (the bull fighting arena they had has been converted to a Shopping mall)
Feel free to PM me if you prefer.
Cheers
Steffen