To inaugurate means to start something, to give it a kick off. The opening day pep rally inaugurates the high school football season.
Inaugurate comes from the Latin word augur, which means taking signs from birds, or telling the future. The Romans would always look for good omens to mark the beginning of something––a war, or a new emperor’s reign––so inaugurate became linked to new beginnings. While inaugurate has long been used in formal contexts, such as the swearing-in ceremony that inaugurates a President's term in office, inaugurate is now also used casually. You might inaugurate your marathon training with a long run.