Will my car insurance cover my medical bills and damages if I'm involved in an accident while driving someone else's vehicle?

It really depends on the state. In general, insurance follows the car. However, if you’re talking about medical payments or Personal Injury Protection (if applicable in your state), those coverages typically follow the driver. So the liability coverage (bodily injury, medical payments, etc.) is going to come from your own policy, and the comprehensive and collision coverage is going to come from the policy insuring that specific car.