When it comes to describing someone who is being productive and actively engaged with work or a task at hand, there are plenty of synonyms for the phrase "gets busy with." A few examples include: "dives into," "tackles," "immersed in," "engaged with," "plunges into," "takes on," "occupied with," "devotes oneself to," and "throws oneself into." Each of these phrases conveys a sense of focus and commitment, suggesting that the person in question is fully invested in what they are doing and determined to see it through to the end. Whatever choice of synonym you choose, they all work well to convey the same idea.