The first wizard appears after you reach Level 11. Completing his tasks will unlock the Wizard's Tower in the Special Shop. Once you've purchased a Wizard's Tower, you'll also be able to purchase an arch wizard (who can also be found in the special shop).
Charging the Wizard's Tower causes wizards and arch wizards to create magic mushrooms and flowers. Charge the tower by dragging magic onto it.
Earn experience points by dragging a magic wand over the magic mushrooms and flowers.