It’s a fine word, which hints of galloping about in frolicsome high spirits. The Oxford English Dictionary’s definition, written in an earlier age, suggests it means “to gad about in a showy fashion”.
I’m not at all sure how often people still describe themselves or others as gadding about (it feels very old-fashioned to me) but you do it by going from place to place in search of entertainment or pleasure, usually with a person of the opposite sex. Its origin in an old and obsolete German word gadling for a vagabond points to its disreputable nature.
Gallivanting is much like gadding about, though ideally you should adopt a more ostentatious or indiscreet demeanour. Its antecedents are rather less clear: the experts wave vaguely in the direction of gallant, meaning a dashing man of fashion, a fine gentleman, or a man who pays special attention to women. That’s from the Old French galant, from galer, to make merry.