What’s the first thing you’d recommend a software leader do when starting a new role in a new industry?

Read The First 90 Days by Michael D. Watkins.

To add a little bit to Paul's response: it's important to demonstrate that you are not a shoot-from-the-hip sort of person (hence Paul's very important advice), but you also want to show that you are not indecisive and passive.

While you are meeting your colleagues and understanding their stories, create a 30-60-90 day action plan, then meet with your manager. Let him know that you have put together a *tentative* action plan (that you will continue to refine as you understand more of the challenges and priorities) but you wanted to make sure you clearly understood from them (your manager) exactly what their expectations are for you during this period.

This will show that, while you are perfectly able to be self-managed and dig straight in, you want to ensure you are aligned with your manager's priorities.

Listen, learn and educate yourself - meet your colleagues and stakeholders, and understand their perspectives and pain points before making important decisions in a potentially very unfamiliar environment and market.