There are a plethora of synonyms for the phrase "get busy with". One could say "engage in", "dive into", "tackle", "start working on", "get to work on", "commence", or "undertake". Other possibilities include "dig in", "plunge into", "immerse oneself in", "take on", "begin", "launch into", or "embark upon". The right phrase will depend on the context and desired tone. However, these options offer a wide variety of ways to express the idea of starting to work on a task or project with intent and focus.