I will be in Spain from March 15-23, visiting Madrid, Valencia, and Barcelona, and was wondering what the weather is generally like during this time period. Can i pack my shorts and swimming trunks or will I need sweaters or even a jacket? HELP!!

All of the above! I'd take one pair of shorts and a sweater for sure. The jacket will come in handy at night and if it rains. You may hit some nice weather but I doubt you'll be going swimming (depends on how thick your skin is)! If you want to soak up some possible rays your shorts will be just fine.

San José: Tuesday, March 19th, is when las Fallas de Valencia happen. Khaki's and a light jacket should be fine - if it's a sports coat (a little dressier) that might even be better. I wouldn't take any shorts. It shouldn't be that warm (although we had a call from Madrid Sunday and they said the weather was spring-like).
From those dates and the length of your stay It sounds like a spring break trip to me. Don't try to crowd too much into one week; just take the clothing you'll definitely need.

I´m in madrid now, just last week people were wearing short sleeves, and now the world once again is bundled up in winter coats and wool scarves. Does anyone know when it should be getting warm for good?

Well, we are still in winter, so is not strange to have cold.
During march, april and even may, anything can happen in madrid from very cold to hot. Spring and Fall are like that here

What about today Carlie Ann? wasn't a wonderfull spring day??