It's Day 3, and Delilah is asking me if I am ready to officially start my job; there's even a nag exclamation point on the right hand edge of the screen. What is the significance of this? Can I wander around and explore indefinitely at this point before beginning the game proper?

You can continue to wander as you please but the narrative will be on hold until you commit. Your explorations are limited to the areas of the map you've already accessed - so if you want to explore new areas you need to get further into the narrative.