Blue coins are not simply coins that are blue; for every ten that you collect, a Shine Sprite is waiting as your reward. There are many blue coins spread out around the game; you'll find them inside zones and out in the streets of Delfino. Take them to the bear in the boathouse (the portal to Ricco Harbor) to buy Shines with them.
Around Delfino
There are many blue coins that cannot be found with specific levels, they are hidden around Delfino Plaza itself.
Fruit lady island
Just offshore there is a small island with a palm tree and a Pianta lady on it. Go to the boathouse in Delfino Plaza (where the portal to Ricco Harbor is) and look out to sea and you'll notice the tiny island. Swim out there and talk to the lady. She'll ask you for 3 coconuts, and as luck would have it, there are 3 coconuts hanging in the tree right next to her. Spray them to knock them down and then put the in her basket one by one. Don't worry if some of them fall in the sea as 3 more will grow back again straight away.
Fruit lady 2
Another fruit lady is waiting in the dock area of Delfino Plaza, outside a shop near the waterfront. She asks for bananas, which can be found at one of the fruit stalls in Bianco Sqaure and on top of a nearby building.
Fruit lady 3
This woman is standing next to the giant Shine Gate in the dock area of Delfino Plaza. Here basket is high off the ground so you'll need to throw the fruits in.
Flaming Pianta
This might sound like the name of a refreshing cocktail, but early on in the game there actually is a Pianta fellow running around the streets on fire! He'll appear after you've collected your first few shines. So, when you see him, make his day and give him a quick squirt. In return he'll give you a blue coin.