Here are my first steps when I take on a new team/department:
Meet the team before you start if possible. Also meet with your manager before you start to understand the lay of the land, find out what the problems are etc. I bagged myself a really amazing project by doing this, looking keen and invested etc.
Read all the HR policies. V important as a manager/leader to know them and then apply them fairly. One of my key messages is always that if you want to know what I'm going to do in a particular situation, read the policy. Fairness is important to people.
Have 1-1s. Prioritise your direct reports, your equivalents in key other teams and your manager's manager. Approach everyone with curiosity.
If your direct reports also have direct reports (and they aren't too numerous!) try and have 1-1s with them in the first couple of months. Be in listening mode and if there are any problems you can solve quickly, do it - eg if someone is having issues getting a special chair etc. You wouldn't believe how far this kind of stuff gets you - showing genuine care for them.
Get the balance between respecting those who already do the job and the fact that you've been hired for a reason. This is why speaking to your upwards chain is key - find out what problems they want you to solve. But don't take too long listening to people before you act if you spot major issues. Remember, though, to bring people along on the journey rather than imposing your will (unless policies and/or the law are being broken of course!).
Accept you can't please everyone!
Remember you were hired for a reason and it's totally normal to feel utterly out of your depth for a while.
Good luck!