Are you hiring?
The first thing I do is get all the factual and confidential details about every contract every player has, and go over the current and future cap implications with the team "cap expert". I may try to pick the brain of the outgoing GMGM as much as possible, and use or discard what I find out. I determine division of duties among subordinate staff, assume ownership of all file systems, keys and passwords and review extent of record keeping. I assign any needed record or document creation tasks to retained support staff with 24-48 hour deadlines. I coordinate best practices with the personal assistant or administrative assistant. I may email the other GMs in the league to introduce myself.
I conduct interviews with all players to hear the what they have to say and to get a feel for the locker room issues/environment. I assess any obvious issues with administrative departments and determine whether changes need to be made there within the next year, and solicit resumes league-wide for a senior consultant. Then I interview the coaches to make similar determinations, and come up with a general hiring/roster plan or goal based on the results. Immediate vs long-term needs are prioritized.
Regarding player moves, in addition to the above info I solicit appropriate staff for information regarding possible short-term trading partners and acquisitions based on real intel rather than message board rumors. Same goes for injury discussions with medical staff. I look at what's coming up in the draft for the next few years and anything else the scouting department has, and of course what we have in the farm system.
That's off the top of my head, and the rest depends on all that.
Playing "I would get this guy and dump that guy" is fun, and it's fine, as long as you realize it's fantasy and wish-listing. Which is why the "what would YOU do" question is never going to get a realistic answer on a message board, and never be a gotcha moment. It's all speculation.