I lived there for 2 years and always recommend Margarita Blue which is at the bottom of the Ramblas on a side street on the left. Get there before you want to eat as you'll need to book a table and then have a drink while you're waiting. It has a nice atmosphere that you feel like you can combine it with a night out and stay in the bar after you've eaten.
The Born area has nice places to drink and eat and is full of trendy locals.
Les Quinze Nits is off the Ramblas and people start queuing quite early for it. But it's a big restaurant so the queue moves quickly once it opens. It's full of locals.
The tourist bus is a good way to get around and it takes you to places you wouldn't bother going to, like Parc Guell, but IMO are worth it. Don't bother going inside Sagrada Familia, just looking outside is enough.
There's a funfair on Tibidabo and the tourist bus takes you to the tram at the bottom of the hill. The market on the Ramblas is worth walking around.
Els Quarter Gats is a café off Placa Catalunya which is where Picasso and his friends hung out in their hey day.
People always go to the new port area but I think it's awful. It's full of stag weekends, hen parties and Brits. There are far nicer areas in Barcelona.
Hope you have a good time. Take care with opening times as it's Semana Santa so there might be some closures.