Using the Pen tool from the tools palette, start tracing the image. The bézier curves may seem difficult to control for new Illustrator users, but it is just a matter of practice to get used to them. Try to add anchor points only where needed, and use as few of them as possible. Your curves will have better form and harmony.
The pen our character is holding is a symmetrical object, so we don’t have to design the whole object. Using the pen tool from the tools palette design half of it. Duplicate the shapes by selecting both shapes and pressing Alt/Option + Shift and drag to the left. Select Object > Transform > Reflect > Vertical Axis. Join the paths by selecting two open-ended points and clicking Object > Path > Join or Ctrl/Cmd + J.
Although difficult to control with a mouse, a Wacom tablet and pen can be extremely useful when using the Pencil tool from the tools palette. Use this tool to draw freeform paths. Use the Smooth tool from the Pencil’s tool expanded palette next to the Pencil tool to smooth any shaky paths.