For recommendations on activities, have a look at these pages:
My favourite museums are:
Tibidabo is one of the hills in Barcelona offering fantastic views of Barcelona (if you saw the Woody Allen movie Vicky Cristina Barcelona, this may ring a bell). You can take an old tram up to the level of the Mirablau bar and walk up to the Tibidabo Cathedral or take the cable car. From there you will also see the old attraction park, which was built in 1889. I never went inside the park but you can see it from the Cathedral. Some pictures here: http://www.barcelona-tourist-guide.com/en/attractions/tibidabo-barcelona.html
The Mirablau bar is perfect to end with a nice drink and a fab view.
The Miro museum or MNAC museum can be easily combined with a visit to Montjuic offering a great view on Barcelona, buildings dating from the Olympic Games. From Montjuic (Mirador), you can take a cable car down to the beach in Barceloneta. On top of Montjuic there is also a castle, which is worth a visit.
There is no shortage of nice restaurants in Barcelona but here are a few tips. Whatever you do, don’t eat at any of the restaurants on the Ramblas. The quality is bad and the prices are ridiculous.
Oleum at the MNAC Museum
This restaurant is inside the National Art Museum and has a magnificent view over Barcelona. It is located on Montjuic. You can walk up from plaza España and take the escalators towards the big palace you will see (it is in that palace). Many times, if you book through restalo or el tenedor, you can get a 30% discount.
Restaurant of the Maritime Museum
The restaurant of the Museo Maritimo offers a nice lunch for 10 EUR on weekdays and they have a very nice terrace.
There are a few nice brunch places in town.
Brunch & Cake, Calle Enric Granados 19
Picnic, Calle Comerç 1
Federal Cafe, Calle Parlament 39
Cerveceria Catalana, Carrer de Mallorca, 236
Set de Born, C/ Esparteria 7 (this place specializes in charcuterie and cheeses, the website does not look super nice but it is a cosy place)
Euskal Etxea, Placeta de Montcada, 1 (this is one of the best basque tapas places in Barcelona. You order a drink and you can just take whatever tapas you fancy from the bar. All the tapas come on a stick. You keep the sticks and you by sticks)
Taller de Tapas, Rambla Cataluña 49 (by metro Diagonal)
This is a very nice tapas place.
Mercado Santa Caterina, Avenida Francesc Cambo 16 (by metro Jaume I and Urquinaona)
Nice restaurant inside the market with a lot of fresh fish. At the bar you can order tapas as well. http://www.cuinessantacaterina.com/