Parts, etc. usually online, although my shop will do an order and get it within 48 hours at a price competitive to many online places.
Cycling clothes always in person locally. The sizing of bicycle clothes is so non-standard that it is not worth the risk, except for gloves. Even then the gloves are cheaper locally.
Tires online, but now that there is a local dealer with Compass/Rene Herse in stock, local.
Helmet and shoes, always local to ensure perfect fit.