The items you pick up or buy in the world will be stored in your inventory. Access your inventory by clicking the hero icon in the bottom left corner, and then change to the inventory tab.
Let's say you have bought that shiny new sword that you always wanted. Now you want to start using it. This is how you do it:
Click the hero icon in the bottom left corner.
The character info screen should open.
Change to the “Inventory” tab.
Here you see a both an overview of what items are currently equipped, and what items you have in your inventory.
Find the item you want to equip in your inventory.
Tap the item you want to equip.
The item info screen should appear.
The item is now equipped and you are ready to start using it. On a side note, you can also long click items in the inventory to equip them more quickly.