Get under way is a phrase that means to start something or get something started. There are several synonyms that can be used for this phrase, such as commence, begin, launch, initiate, start, and kick off. Commence is a formal way of saying to start something, while begin is a more informal way of stating the same thing. If you're trying to convey a sense of grandeur in your writing, launch may be a good synonym to use, while initiate is similar in meaning but has a more technical connotation. Start is a general term that could be used in various contexts, and kick off is a more casual way of saying to get something started.