What happens to a task when I assign it? Am I still able to see it in my "Today" list? Do I get a notification if the person completes it?
If you assign a task, it will not show up under your "Todays" list anymore. As for notifications, as long as they're enabled, you'll get a notification when they complete it. Here's a full list of notification (email and push notifications):
Comments attached to me
Tasks assigned to me
Collaborator joined project
Collaborator declined invitation to project
Collaborator left project
Collaborator removed from project