Awesome trip. We did this for 10 days over Christmas break. Our trip was almost similar to your itinerary.
If you are vegetarian like me, remember the national vegetable of Spain is the Iberian Ham ;-) If so, get the happy cow app on your phone and you can find great places. Hot chocolate and churros are a great mid-afternoon snack. Make sure you get the freshly squeezed orange juice that every place has. And coffee everywhere is awesome.
Start off in Madrid and end in Barcelona. So your trip is not exactly round trip, flying into Madrid and flying out of Barcelona.
In Madrid, make a day trip to Toledo. Awesome little city. You can book tickets locally and you can do this when you are there, so you can pick a good day to go there since you will have a lot of walking within Toledo. Also, make advance reservations for the Prado museum, much before you leave for Spain.
Madrid to Seville, take a train. Nice to see the countryside and trains are awesome.
Seveille to Granada, also take a train. Make sure that you book tickets for the Alhambara much in advance. Again, book this much in advance of your trip to Spain.
But Granada to Barcelona, take the plane. I am not much into cathedrals, but the Sagrada Familia blew me away. Gaudi was a genius.
Get a phone with data for your trip. Google FI will work, we took an old iPhone and bought a SIM card there. Just take your passport when you need to get the SIM card since you need id for that. We used taxis when we were there, but with a phone, you can use Uber.