I recommend 4 full days in Barcelona alone to do it full justice (and this does not include Parc Guell)
DAY 1 Ramblas & Bari Gotico
Metro to Placa de Catalunya, Barcelona’s main square
Explore the Ramblas from top to bottom taking side diversions
On the right C. d’Elisabets to the museums MACBA and CCCB (check for current exhibits)
Further down to La Boqueria markets
Left side thru the Gothic Quarter via Placa del Pi & Placa de Sant Felip Neri to the Cathedral
Placa del Rei and the wide Placa Sant Jaume flanked by official buildings of the city and state
Return to the Ramblas (at the Liceu opera house) down the shopping street C. De Ferran
Further a little to Placa Reial, harmonious enclosed square with palms trees & Gaudi lanterns
And then Palau Guell (Gaudi) off on the right
Finally the port and the column with Christopher Columbus gesturing towards the new world
Metro Drassanes is close by
DAY 2 Montjuic & Olympics
Metro to Placa d’Espanya
Les Arenes, once the bullring, now housing shops & restaurants (fighting is banned in BCN)
2 large towers opposite and the Av de la Reina Marina Cristina towards the Palau Nacional
This area of exhibition space was the site of a Universal Expo in 1929
(Barcelona had Olympic visions way back then but the Spanish Civil War put them on hold)
Caixa Forum and Mies van der Rohe Pavilion are both worth a look
Escalators to the Palau and the MNAC museum, well worthy of a visit
The Anella Olimpica (Stadium, Palau St Jordi, Communications Tower) is laid out behind
Now back down the hill to the excellent Miro Museum (Fundacio Miro)
Piscina Municipal close by – spectacular Olympic diving venue
Along Av. Miramar to the Funicular connecting with Metro at Parallel
DAY 3 Gaudi & Modernism
Metro to Sant Pau to explore the modernist hospital complex of Sant Pau
Walk down the Av de Gaudi to Sagrada Familia (Gaudi) – PREBOOK TICKETS !!
Metro to Diagonal and Passeig Gracia for Casa Mila (Gaudi)
Walk down Barcelona's majestic boulevard to crazy colourful Casa Batllo (Gaudi)
Casa Amatller is next door and the Loewe store on the corner is in Casa Lleo i Morera
Finish at Gran Via – Metro Passeig de Gracia is closeby
DAY 4 El Born & Barceloneta
Metro to Arc de Triomf
10 min walk to Palau de la Musica Catalana – PREBOOK TICKETS !!
Through the revitalised area of El Born to the Picasso Museum – PREBOOK TICKETS !!
On to Santa Maria del Mar and then into the Parc de la Ciutedella for a relaxing circuit
Past the historic railway station, Estacio de Franca to the old fisherman’s village, Barceloneta
The ‘D’ shaped W Hotel stands at the port entrance
Walk the other way along the beachfront towards Frank Gehry’s glimmering ‘Fish’ sculpture
Finish at the Hotel Arts and Port Olimpic – Metro Ciutedella / Vila Olimpica is close by