Using the shapes tool is a good way to start, in my opinion. It allows you to easily create shapes with perfect curves, the correct dimensions and the proper proportions. Seeing as I like to work in an exact way, I personally use them as well quite regularly. By merely adding and subtracting shapes (and by rotating and skewing them), you can get quite far
While Illustrator may be more powerful, creating vectors is actually easier to do in Illustrator than it is in Photoshop. On the first day that I used Illustrator, without any prior experience with vectors, I was already able to create my own avatar as a vector. I did so only by adding and subtracting shapes in order to create a perfect final shape. Other logos, I had to do freehand seeing as only using shapes would have been nearly impossible, this one for instance:
[link] When using Illustrator, you should add the Pathfinder toolbox to your workspace though, makes life much easier in my opinion. Of course, you can always ask me if you have questions
You may have probably already seen this tutorial btw, but in case you haven't it's well worth a read:
[link]