Straw hats need special care because of their lightness. The crowns are creased, and blocked or shaped to a particular style, making them brittle. So the more you handle the crown, the sooner it will dry out, crease, or even crack. The straw should be handled by the brim, which is more flexible and easier to handle. That way the crown will stay clean, last longer, and look nicer.
The straw can be dusted with a whisk broom and stains can be wiped off with a damp sponge. If you get caught in the rain with your straw, the hat will be more pliable. So don't just throw it in the corner to dry out. Instead, turn the leather sweat band all the way out, turn the brim up, and set it on the sweat band to dry, away from heat and out of the sun, in the shape you want to find it the next day.
Also, after a particularly hot, sweaty day, turn your sweat band out so it dries faster. When you aren't wearing your hat, storing it properly is important. The hat should be stored in a box, or covered, and set on its crown. That way it will stay in the shape you want it in.