Santander is 64 miles from Bilbao, so a very, very costly taxi ride away. A bus would be perfectly viable. There is also train service, but is much slower.
I loved my ten days in Barcelona last August (not counting time in Girona) so I heartily approve of the time you plan to spend there.
Off the top of my head, these are the places for which you should buy advance tickets (if you want to see these sites, I mean) to avoid standing in probably-lengthy lines:
- La Sagrada Familia: separate tickets for church and tower. Usual recommendation is to buy the first morning time-slot for the church so you have some time before it is crowded. If ascending a tower, schedule that perhaps 1 hour later; you can spend more time on the ground floor of the church afterward if you want to. The tourist office told me buying the ticket 2 days ahead would probably be safe, but all bets are off if there's some huge congress going on while you're there. You might need 2-1/2 hours for LSF. There's a small museum and a nice gift shop.
Parc Guell: Again, early morning is recommended. If you want to see the gatehouse (which Rick doesn't particularly emphasize and which I didn't see), you will definitely want that first time slot and should go directly to the gatehouse. There's a separate line there which builds up rapidly as soon as the park opens. This is an outdoor site, so if it's hot, that's another reason to prefer an early-morning visit. The tourist office told me buying 1 day ahead should be OK. Takes time to get here if you don't taxi. Bus is better than Metro because the latter leaves you about a 1-mile uphill walk from the park.
Palau de la Musica Catalana: Requires a tour (English available). Tours do sell out, but you can certainly take care of this after you arrive. This visit takes less than an hour, all-in.
Picasso Museum. An utter zoo. Don't think of doing this without pre-purchasing a ticket. If you don't like early Picasso or Cubist Picasso, I'd skip it; it's not like there aren't plenty of other good museums in Barcelona.
Casa Mila/La Pedrera. Buy a ticket in advance to avoid the line Can do this once you get to Barcelona; I'd be surprised if any day every sells out completely until late in the day. Someone pointed out the other day that you can pay a bit extra for a sort of wildcard ticket that doesn't tie you to a specific time slot. Might be worth it for the flexibility, but the site will be crowded whenever you go. Rough estimate: 90 minutes here if you ignore the interesting information on the top floor about other Gaudi sites; otherwise, perhaps 3 hours.
Casa Batllo: As above as far as tickets are concerned. Even more crowded. Maybe 90 minutes if you can push your way through the mob.
Places I enjoyed that did not appear to require advance-purchase tickets when I visited:
Sant Pau modernista site (wonderful multi-building site, not yet discovered by masses of tourists. See it now.)
Miro Museum on Montjuic.
Museum Nacional d'Art de Catalunya on Montjuic. Lots of good stuff, not just paintings.
In sum: As long as your visit is, you really don't need to buy any of the sightseeing tickets before you get to Barcelona, but if there are places you want to go on the Day 2, you might want to buy a ticket for at least the first one from home. In 2016 the tourist office beneath Pl. Catalunya was selling tickets to all the advance-purchase places I highlighted above except the Palau de la Musica Catalana and the La Sagrada Familia towers (T.O. does sell church tickets). There's a 1-euro service fee per ticket. T.O. is busy but efficient; very helpful. Lots of English-language brochures displayed.