Before you can begin fishing, you must have completed
The Angler Alliance quest from Barton at Gildon Bridge

The club Membership Card you bought from Bruno doesn't
need to be kept and can be sold to Bruno for 25 wealth

Buy the Angler Vest from Bruno in East London for 80 wealth

Buy the Angler Shorts from Bruno for 80 wealth

Buy the Angler Boots from Bruno for 20 wealth

Keep all three clothing items in your inventory and you will
automatically change into them whenever you want to fish

Stand on a bridge or beside water and press the A button

Wait until an exclamation appears then quickly press the A button

This free video game walkthrough is for the Nintendo DS

Professor Layton's London Life Walkthrough

Professor Layton and the Last Specter

Professor Layton and the Specter's Call