If I was you I would end my trip in Madrid instead of starting it. Your last leg of your trip, Granada to Madrid is a five - six hour trip by bus or train. I do not know how long if you had a car because I never did it.
You can start your trip in Sevilla. If you are fying into Madrid from the US, you can either fly since you are already at the airport or take the AVE train to Sevilla. Then a bus to Granada from Sevilla and end your trip in Madrid and fly home. I recommend not to take the bus from Granada to Madrid, I did it and it was the worst bus trip I ever had. So take the train if you do not want to drive.
Since it is a short stay, Toledo is an excellent day trip from Madrid. Very easy to get to by bus.
March is just around the corner.
AddedI just saw your other message, you only have a week. Skipped Granada and Sevilla. Spend the week in Madrid and do day trips. You can visit Sevilla, AVE is like three hours but do you really want to spend six hours on a train for such a stay in Spain.
Bill