Adobe Photoshop and Adobe Illustrator are the basics of the graphic artist's toolbox. Both are important, but they are different. Photoshop is a *painting* program, Illustrator is a *drawing* program. Photoshop is easy to get started on, Illustrator is difficult to get started. I like both programs, but I am especially fond of Adobe Illustrator. If you like to draw, you should be using Adobe Illustrator.
Please don't get me wrong - I love Photoshop - and use it every day, but many times it is not the right tool for what you need to do. If you are doing an image with solid masses of color, like the logo pictured, you could do it in either Photoshop or Illustrator. If you've never learned Illustrator, I can understand why you would use Photoshop. But Photoshop is the wrong tool for this. Using Photoshop here is like using a paintbrush to do a drawing. You need a pen.
The best place to start with Adobe Illustrator is basic shapes - squares, circles, etc. Once it *clicks* with you that you are drawing, not painting, Illustrator makes sense.