You are trying to see a lot in a short time and the distances are pretty significant in some cases. I recall that it takes more than 2 hours by train from Barcelona to Valencia. And Valencia is not well connected to Sevilla. Also remember that even a 2 hour train ride is the equivalent of losing a half day of touring since you have to get to the station, get to your hotel, check in and orient yourself. You are losing at least a full day and a half in travel with this itinerary.
For Barcelona, which is the outlier in this trip, Gaudi is the big attraction. Sagrada Familia is a must even if architecture doesn't interest you that much. I'd also choose either Casa Mila or Casa Batllo to visit (or both but they each are very expensive). In a short trip, I'd skip Parc Guell. Reserve Sagrada Familia tickets in advance as the lines are very long and they sell out a day or two in advance. Exploring the Bari Gothic and harborfront neighborhoods is worthwhile and if you like modern art, there are some good museums (Picasso, Miro).
Valencia can be one night on such a short trip (depending on your actual travel itinerary). The main boulevard is delightful and you must try good paella there (find a good restaurant for it, not just a reheat from frozen place). If you like wild modern architecture, take the tram out to the Museum of Sciences complex, all designed by Santiago Calatrava. You don't have to visit the museums inside, but the complex is pretty stunning. Valencia doesn't have the big ticket attractions that the other cities have, but it is one of our favorite Spanish cities.
Granada probably should be on your list, but again, how you get there is the challenge. Perhaps fly from Valencia to Granada. The Alhambra is amazing and not to be missed. You must reserve in advance to have any chance of seeing it.
Sevilla is about the Cathedral and Alcazar. Both can be seen in a single long day, or better to split them up. The alcazar can take several hours if you don't rush through. And then plan for a flamenco show at night and lots of yummy tapas.
Cordoba is all about the Mezquita and wandering the old quarter. Easily a day trip or stop on your way to Madrid. There is luggage storage at the bus station across from the train station.
Madrid may be a let down after the older, more architecturally and historically rich cities. But the art is top notch - the Prado for half a day and modern art at Reina Sofia and a smaller collection at the Theissan.
