To equip your fairy, simply drag it into the second slot of your Inventory.
Then you could find your little partner flying around you with its status bar under your character status bar.
When your fairy is hungry (SP is not full), You could feed it with pet food or rations.
To feed your fairy: double click the pet food or ration, a chicken leg will show up, move it onto your fairy, then click.
You could level up your fairies with energy food. The operation is similar to Raising.
You could teach your fairies with Skill books. The operation is similar to Raising.