Wool Items: hand wash in cold water and lay flat to dry (to avoid shrinkage and felting)
Knit Items (cotton or synthetic): machine wash cold and lay flat to dry (to keep its shape)
Silk Items: hand wash in cold water, hang to dry (to avoid damaging)
Cotton: machine or hand wash cool water, hang to dry (to avoid shrinkage and bleeding)
When in doubt, the best bet is to always hand wash in cool water and air dry!
Some items can't be washed in water or are too delicate and need to be washed by a professional dry cleaner.
Many vintage items are best to be dry cleaned as well, if the fabric seems delicate from age.
Beware that moths love to eat your wool and silk fabrics! Before you put any clothes away for storage make sure they are freshly cleaned, this will get rid of any eggs or dirt which attract the bugs. You can also find red cedar blocks or wood chips which help to kill moth larvae. Also storing your woolens in cotton bags is helpful as the moths don't eat through cotton.
Knit items such as sweaters should always be folded and NOT hung up in your closet. Hanging your knit items will cause them to distort in shape, especially the shoulder area.
A really great resource for more specific laundry advice is the Martha Stewert website. https://www.marthastewart.com/1537276/laundry-linens