Hi,
Yeah, the traffic in Barcelona isn't that great- but the public transportation there is extremely efficient, especially the metro. Valencia isn't too big and you can get across it by bus in about half an hour- I did it. The buses to more remote places take longer, though, and you might just be dropped off by the side of the road once you get to the town- that happened to me when my friends and I were going to Cullera.
My advice, though, would be to stick to Barcelona and maybe take a daytrip out to one place. Five cities in nine days is WAAAY too much! You can easily just spend those nine days in Barcelona alone! There is so much to do and see there! I would recommend seeing Gaudi's archetecture when you are there- it is absolutely out of this world, like nothing you will see anywhere else. I tried to see Barcelona in 3 days and regret it to this day, so to barely skip over it in a five city, nine day wirlwind would be a mistake for sure.
Have fun in Spain! :)