Where is a colostomy usually done? At a hospital?

Always. Creation of a colostomy requires general anesthesia and, depending on the indications for surgery and what else is being done, an in-patient stay in the hospital.
Absolutely. A colostomy must be done in an operating room under appropriate anesthesia. Otherwise, doing it in the comfort of your living room may be a bit painful. It entails a surgeon pulling up a loop or limb of small but typically large intestine up to the surface of your abdominal skin, going through all the musculature and sewing it in place. A colostomy appliance is then attached to collect the stool.