Sort this out and the rest is easy.
Take the train in from the airport
Skip the taxi queue. The train from the airport is very comfortable and it drops you at Passeig de Gràcia, which is city centre — you walk out of the station into the middle of everything.
Line R2 Nord from Aeroport station. The station is attached to Terminal 2; from Terminal 1 take the free shuttle bus, about ten minutes. Roughly half an hour to Passeig de Gràcia.
Open in Maps
Buy a T-casual before you do anything else
Ten journeys on one card, and far cheaper than buying singles. It works on the metro, the buses, the tram and the local Rodalies trains. It cannot be shared between two people travelling together, so buy one each.
Any metro machine sells them. Trips further out, Sitges especially, may need extra zones — enter the destination and the machine works it out. Fares change, so treat any price you read online as a guess.