In Breakwater Beach is a sunken broken boat in the water, Ghost Ray it and you will see the Fishing Pole at the bottom of the boat. When you click on it, it will make you play a game of fishing and if you win the game you get to keep the Fishing Pole. If you don't win you will have to try again.
Your Fishing Pole will appear in your backpack under Loot. Click on it and it will be in your hands (you can only cast it in the water).
You can either find a Wild Fish, a junk item (see article Quill the Junk Vendor for junk items) or a school of fish for a game of fishing. To find a Wild Fish you must be fishing at Breakwater Beach or Tanglevine Jungle. Wild Fish will run away when you leave the area (you have to catch them again) and when you put them in free play they run away to another place in the area you are in.