Do something in a careful or cautious way, and you'll do it safely. The most important thing you'll learn in a drivers ed class is to drive safely.
"Travel safely!" you might call to a departing friend. You're encouraging them to avoid danger as they make their way home. You can bike safely by wearing a helmet and using hand signals, and use power tools safely by following all the rules and focusing on your work. Safely comes from safe and its Latin root salvus, "uninjured or in good health."