To me, the first 90 days are all about communication with your new team and cross-functional partners – even more so if you're setting up the Product Management function for the first time. Treat your communications a bit like internal marketing for PM (and yourself), meaning they should be tailored to your audiences, showcase how your PM skills create value for the team, and address potential concerns head-on.
The first 30 days are your time to listen and learn. Spend time with Engineering and Design to observe team dynamics and collaboration styles. Also talk with Marketing, Sales, Data, and Operations, to align on what they expect of you – all the while being aware that if they have not worked with PMs before, you may have to inform them what to expect of you.
At the 30 day mark, share back in writing what you heard – to demonstrate that you listened and show that you're integrating the information that you've been given.
At 60 days, be ready to publish your hypotheses about product strategy and thoughts on what's needed to validate those hypotheses. Depending on your background, you may not feel like an expert in this space yet, so I'd encourage you to be vulnerable and ask for feedback even after the memo is out.
By 90 days, you should have a plan of action that is fully aligned with Engineering, Design, and cross-functional partners. You should be a part of the team's cadence and active in all the right meetings. You should also aim to deliver frequent updates – bringing everyone along on your journey, sharing what you're doing, and tying these to long-term shared goals.