You can carry 16 items at a time, but with the essentials (shovel, fishing pole, net, and wetsuit) on hand, your carrying capacity is greatly reduced. Fortunately, wearing and holding some items will free up some much needed inventory space. Also, by attaching items to letters, it's possible to hold even more, so fill your pockets with letters before you head out.
Equipping the shovel, fishing pole, or net removes the item from you inventory and puts it in your hands, freeing up an item slot. Even when you go inside a building and watch your character put the tool away, it won't go back into your inventory (don't worry - the item doesn't glitch away). Wearing the wetsuit also removes it from your inventory. Wear it while holding a tool, and you'll have two extra inventory slots to play with.