Hi Helen
You will need to learn how to use the Pen tool and manipulate the anchor points. Its not the easiest tool to use and will take some practice to get used to.
Have a look at Illustrator’s Pen Tool: The Comprehensive Guide | Vectortuts+ as a quick beginner guide.
That’s a very complex image to use to learn Illustrator tools. You should first master the basics with simpler images. I’m not saying that you can’t achieve the same effect on your first try, maybe you can. Besides understanding the Pen tool, you might find the “Pathfinder” tools helpful “Add, Subtract and Intersect”. You’ll also need to learn basic gradients settings to achieve similar shading.
Personally I wouldn’t bother if it is just for making a logo for your company. It represents a lot of work to get looking professional. Especially if you are going to use it in a business capacity - I’d recommend paying some nominal fee to download a vector image from any of the decent image libraries there are online now. Alternatively you could look for free image sets that could give you a head start to adapt into a logo for your company.
I guess it depends on time and resources, if you have some skills already then learning illustrator and the pen tool etc will never be wasted. But there are good sources of vector images now that could save you a lot of time and trouble.