Flying into Barcelona and out of Madrid--Recommendations?

The issue is MAD to BCN is on a high-speed line, so the most efficient thing is to take that between them, unless you want to stop in Zaragoza. So, what a lot of people do is plan day trips from both cities, or you can do side trip excursions from both cities, but you will not necessarily be stopping in between. To go to Granada, you will pass through Madrid if you go by train (some fly).
With day trips, those three would be plenty for 13 days, but depending on the pace you enjoy, you could add a stop in Sevilla, Cordoba. I find it can get overwhelming if you try to pair Andalucia with Barcelona, but it is done. The options are dizzying at this stage, but once you do some reading, something should rise to the top as most compelling to you. Good luck!

Why are you going from Barcelona to Madrid, then leaving for other cities and ending up in Madrid. That's not very efficient. I would probably fly to Granada from Barcelona, then train to Sevilla and end up in Madrid to spend the last several days.