When you first begin your adventure, your character will only come equipped with two different Weapons: a Cutlass and a Pistol. You can check out the Weapon Shop in hopes of buying better weapons, but they are pretty darn expensive and your basically broke when your start your journey.
Believe it or not, however, you actually can add two additional Weapons to your arsenal immediately after beginning the game. In order to do so, simply go to the locker outside of the Weapon Shop or near your Captain's Table (on your Ship). Here, you can access your armory and actually swap either your Cutlass or Pistol for one of two other weapons!