Doing a 5-week Iberian peninsula mostly in Spain (Barcelona, San Sebastian, Madrid, and Seville) and Lisbon for 9 final nights. Starting in Barcelona (6 nights) in an apartment 2 minutes walk from Paral.lel metro station.
So let's see if this is doable. We want to see the following but not necessarily in order written:

Gaudí’s Sagrada Família & Park Güell
Montjuïc Hill via the Montjuïc cable car and Joan Miró Foundation
Gaudi's Casa Milà & Casa Batlló & Casa Museu Gaudí
Barri Gòtic (Gothic Quarter) including Plaça del Rei, Mercat de la Boquería, Las Ramblas ending at Columbus Monument
Picasso Museum & Quadrat d'Or
Cathedral of Barcelona & Basilica of Santa Maria del Mar
Palau de la Música Catalana (Palace of Catalan Music)
Museu Nacional d'Art de Catalunya
And of course doing the Cava/Vermot crawl every evening!

And if we have time left, will try to do:
Park of the Labyrinth of Horta (about 30 minute metro ride north of our neighborhood).
Girona and Monastery of Santa Maria de Ripoll where Wilfred the Hairy is buried (40 minutes train ride from Barcelona-Sants)

Doable? Any suggestions? Additions?