Use 'wary' when you want to express caution for personal safety or to avoid problems. Parents teach their children to be wary of talking to unknown people.
Pro Tip 2/3
Alert and Ready
Being 'wary' often implies being alert and ready to respond if something goes wrong. The campers were wary of wildlife while hiking in bear country.
Pro Tip 3/3
Evaluate Risks
'Wary' can indicate that a person is assessing risks before taking an action or making a decision. He was wary about investing in a new company without proper research.