Sorry; I did read the subject line but immediately forgot, and then I didn't scroll back far enough!
So not the best time of year for places like San Sebastian and the Picos de Europa. Barcelona and much of Catalunya should be nice, plus Andalucía. The area around Madrid will be somewhat cooler.
The folks really familiar with the various possibilities will comment soon, I hope. My interests are so different from yours that I can't be of much help.
I'll just warn you that the Alhambra sells out very early, so if you're interested in Granada (from which you might prefer to fly back to Barcelona), you should start your ticket search as soon as you have dates to look for. My guess is that the basic tickets for the entire month of April are sold out, but you might snag a Dobla de Oro ticket (covers more, costs more), or you might be able to get a ticket via the Granada Card (covers even more, costs even more, but it's not ridiculous). It's the time slot for the Nasrid Palaces that is the problem. You can tell whether there are any openings available via the Granada Card before you complete the purchase. If no luck via either of those methods, you may be able to get a ticket through your Granada hotel (probably at some extra cost). In all cases, the earlier you try to accomplish this, the better your chances.
The Alhambra is not the only thing to see in Granada, but your time is limited and could easily be filled with other destinations (Toledo, Segovia, Cordoba, Seville as well as Madrid and Barcelona--and those are just the larger places). I'd prefer to stick to just three cities if you choose largish ones (Madrid, Barcelona, Seville). Or maybe two if you decide to include a lot of enticing outdoor activities and/or side-trips to small towns.
Barcelona has a lot of top attractions that tend to be open every day. I think that city won't be too affected by Semana Santa, but since you're flying home from Barcelona, you will probably be there after Easter (I'm guessing) unless you decide to make this a mainly Catalunya trip--which is definitely a possibility. On the other hand, Barcelona is very heavily touristed and might be even worse than average during Easter Week. It's a place where you often need to buy sightseeing tickets in advance.