Sort this out and the rest is easy.
Take the train in from the airport
Skip the taxi queue. The train’s comfortable and it drops you at Passeig de Gràcia, dead centre. You walk out of the station into the middle of everything.
Line R2 Nord from Aeroport station, attached to Terminal 2. From Terminal 1 take the free shuttle bus, about 10 minutes. 30 minutes to Passeig de Gràcia.
Open in Maps
Buy a T-casual before you do anything else
This is the old T-10 under a new name. It replaced it in 2020, and the catch is that the T-10 could be passed round a group and this one can’t. 10 journeys, 1 person, so buy one each. Works on the metro, the buses, the tram and the local Rodalies trains.
Any metro machine sells them, around €13. Each journey allows 75 minutes of changes, so a bus onto a metro is 1 trip, not 2. Trips further out, Sitges especially, need extra zones; enter the destination and the machine works it out. Fares change, so treat any price you read online as a guess.