There are plenty of different synonyms for "moving quickly," depending on the context. Some more formal synonyms include "hastening" or "scampering." More casual synonyms might include "booking it," "hustling," or "jetting." For situations that involve athletic activity, words like "sprinting," "dashing," or "racing" might be appropriate. When it comes to animals moving quickly, you might use words like "galloping," "bounding," or "darting." There are also plenty of figurative ways to describe someone moving quickly, like "whisking away," "zipping by," or "speeding towards." Overall, the best synonym for "moving quickly" will depend on what kind of situation you're trying to describe.