I will be spending approximately four days in Barcelona (ending on July 10) and then I have about five days to spend in Granada (to see the Alhambra) and Madrid. I fly out of Madrid on July 15. What is the best way for me to go? Should I go straight from Barcelona to Granada (with a night in Granada) and time to see the Alhambra the following day (and a night visit to Alhambra the night I arrive), and then go to Madrid, or should I go from Barcelona to Madrid and then do a round-trip from Madrid to Granada? What is the best transportation route and methods? I am interested in saving transportation time and also being reasonably economical. Thanks for any input.

To best use your time and money I would recommend taking the night train from Barcelona to Granada. That way you don't have to worry about a hotel/hostal that evening. If you pay for a sleeper supplement you should be able to get some good rest then you would have the entire day to go visit the Alhambra and whatever else you want to see in Granada. Then you can take the bus or the train to Madrid the next day/ evening.