How do I test an extension cord for continuity and proper grounding with a multimeter?

Set your multimeter to resistance or continuity mode. For continuity, touch one probe to a prong on the male end and the other to the matching slot on the female end — you want near-zero resistance for hot, neutral, and ground. For grounding, test between the ground prong and ground slot (near-zero resistance), then between ground and hot/neutral (infinite resistance). If you get infinite resistance on a conductor, the wire is broken; if you get continuity between ground and hot/neutral, there’s a dangerous short. Also test any built-in fuse or circuit breaker separately.