Shorts are worn in the summer months of June, July, and August. Locals also wear shorts in late spring and early fall if it's hot. Even if you get a random hot day in the winter, Spaniards will not wear shorts.
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The dress code is in line with that of the rest of Europe: smart, dark-coloured suits for both men and women, although women are not expected to wear jackets. Conservative attire is respected with nothing too showy. Elegance and style are preferable to glamour and ostentatiousness.
European style shorts are becoming far more common for locals in northern Europe, although still not common in France and Italy in particular. But still, on the whole, Europeans tend to dress more formally than we do. Shorts are something you mostly see in a beach resort.
In general it seems a bit more relaxed than in Italy. I've seen locals break the rules while wearing sleeveless dresses (not super short but a couple inches above the knee). The main hangup seems to be around shorts and tank tops on anyone. However, the official rule is shoulders and knees covered.
They will even keep you from going into churches if you are not attired appropriately. Loose dresses meet any entrance requirements, will not earn you stares, and will keep you cool under a hot sun. In Spain, wear a skirt or long pants when touring cathedrals and other religious sites. Definitely no shorts!
Scarves, gloves, and hats are needed throughout many days. From Madrid down, just wear layers. Most days you may need to take off some layers from 11 am to 5 pm. After all, Spain is the country with the most days of sun in the world.