I recently did Everything Wrong With Phantom of Venice. I figured I might as well try a little to plan the speedrun for the game, since it's fresh in my memory.
The game forces you to talk to Colin at the start. Then you go outside, where Nancy runs into Helena. Read the letter that's for Nancy there, then head off to the Piazza San Marco, to pick up Nancy's spy gear.
I have no idea how to navigate the game map in a speedy manner. Any suggestions are welcome.
You want to buy birdseed while you're at the bank. Sadly, you can't buy what you need from the kiosk yet. The first time you go to the kiosk, Nancy says a short sentence about it. Take a look at the sunglasses display here. The pair of sunglasses that Nancy buys later on will disappear, while she's talking, then reappear when she stops. How odd! Too bad I didn't notice that for "Everything Wrong With"!
At the ATM, you put in your card. You can avoid reading the dossier by clicking instructions, next, and then home. You want to return to the Ca' after this, but make a quick stop at Campo Del Frari. Try opening the door, and Nancy notes she needs a bobby pin. This triggers your ability to get a bobby pin from the Ca'; this is probably the ideal time to trigger that.
There's nothing else you can do at this point in the game, so you go back to the Ca'. Simple enough.