Wool and silk are both antimicrobial fabrics that do not need to be laundered with each wear.
I recommend dry cleaning every 4-5 wears, or longer if it suits you. For in-between care, spot-clean mild stains before storing. Steam is safe to use on both the self and lining layers. It will remove any wrinkles as well as deodorize the fabric.
Should you want to press the garment, press on medium-high heat with a pressing cloth between the iron and fabric. Pressing directly on high heat can damage the fabric leaving unwanted sheen.