How can I map out what to do in the first 90 days in a new role (staff and senior staff) and a new job (given factors like new codebase, new processes, people, and work ethics), especially when the domain and coding language are also new?
I talk about this a lot in my Senior to Staff course (and I'll make an entire onboarding course soon!).
One thing I can recommend is what I call the "Talk & Observe" framework. The idea is to talk to a bunch of people on the team to get ideas of what's important, and then observe their behavior if it matches what they claim.
Staff engineers are tricky as that's when archetypes really solidify and the value you add is specialized. The broader all-encompassing advice is to have that honest dialog with your manager about expectations and create an onboarding plan. How engineering teams usually work is that the Staff+ engineers are the "lieutenants" of the engineering manager (EM), so it's obviously in the EM's best interest to set expectations and clarify their vision for the Staff+ engineers to carry out.
In terms of specifics, you can find concrete behaviors expected from Staff+ engineers here: "For a Staff engineer at a Big Tech company, what should their first 3-6 months look like?"
Back at Meta, every new Staff engineer I worked with made it clear within their first 3 months why they had a Staff title. 90 days is a long time, and it's certainly long enough to find a hard technical or organizational problem to solve (expected of Staff+ engineers).