What’s the fastest way to see all tasks assigned to me, or a specific team member?

The easiest way is to click the filters funnel icon and select Assigned to me, or Assigned to and pick a user from the list.

You can also find task assignments manually by typing @ and user initials - e.g., @XY, into either the filters or search box, both located in the top-right corner of any Kanban board.
Filters show results for the open board, while search scans across all boards you have access to.