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Day one, if you planned to walk from Casa Vicens to Passeig de Gracia* there is a nice tapas place called Barra Alta on Laforja 11. It hasn't rained for awhile here I wouldn't worry too much about the rain, the rain forecast is often incorrect! If it does it may pour for 10 mins or so and then everything is dry and dusty again.
As for tain to Tarragona you can check the fares and schedules on Renfe site, it's a short regional train ride which I imagine it is ok to rock up and buy a ticket, nonetheless if you are sure about going then buy the ticket in advance.
One thing to know is that most locals in Barna go away for August because it is hot and unbearably humid, businesses in the non-tourist area are closed or have limited opening hours.
Buy the Sagrada Familia ticket beforehand, you don't want to stand in a long queue under the hot sun.
If you are feeling brave, venture to Gracia's Gasterea at Verdi 39 for dinner, cash only. Otherwise Bar del Pla on Montcada 2 in El Born is my favourite, booking is a must.
Spanish is widely spoken in Barna whereas Catalan is more widely spoken in regional towns. Put it this way, in Catalunya the majority of Catalan speakers speak Spanish but not all Spanish speakers speak Catalan. Continue with your Duolingo because English is not widely spoken here outside of tourist area.
Download the NextBus Barcelona app if you intend to use the bus, click on the individual lines for bus route in map view. Buses are more convenient than trains and less pickpockets too. Last but not least do watch out for your personal belongings, be wary of strangers coming too close to you, and watch out for your phones if you have them on the dining table when strangers approach, many phones disappear that way.
Get this and that's it https://play.google.com/store/apps/details?id=cat.ereza.properbusbcn&hl=ca
If you are here in the beginning of June there is the Primavera Sound music festival, someone already mentioned Sonar.
It was pretty easy to fly to Paris Orly with Vueling.
Next Bus is very handy.