How do you navigate your first days in a new finance leadership role? How does this differ if you're an internal promotion or external hire?

Build relationships and learn the systems and processes as quickly as possible as a new hire in a leadership role.  As an internal for an existing role, adapt quickly into taking on the review tasks and oversight activities of your predecessor to set new expectations and tweak existing relationships and, in the case of internal hire to a new role, move very quickly to establish the differences between the new role and the old and clarify for the immediate team and other key stakeholders through multiple communication avenues.

Understand the team vision, how the team fits to the strategy, what is the expectation. From there, map your stakeholders, understand their basic needs. Of course, connect with your teams, get to know them and their stories and backgrounds, before jumping into work. Building trust, showing humbleness and collaboration are very important steps to achieve success.
If you are an external hire, I would add learning the company's way of operating. Is it a process-oriented or relationship-oriented environment? That will help you understand how to navigate and build your credibility and brand. Don't try at first to bring your external perspective. Listen seeking for understanding.