I would like to know that which tool is best for illustration for my art designing. Actually I am confused between Photoshop and Illustrator.
Illustration isn't a style or drawing method. It means that a drawing or painting is used to supplement something else, usually a writing.
For example I believe this wasn't originally an illustration:
But it can well be an illustration in a book of renaissance art.
I believe the next can be sold as an independent piece of art only if it has a finely handcrafted wooden frame:
But it surely fits well to some math writing.
Illustrations as well as independent pieces of art can still be made with no software tools like it has been hundreds of years.
If you want to make images with software, you need different tools to different purposes. Photoshop is good when you want to imitate painting or manipulate photos. For painting only specialized painting programs are clearly more flexible. In Photoshop you can also draw sharp edged lines and colored areas and insert texts, but vector drawing programs such as Illustrator have much more flexible tools for it.
Best in vectors is their free scalability without increasing blur nor pixelation. Photoshop makes raster (=bitmap) images and one must be aware of their limited scalability altough you can inside Photoshop keep texts and drawn paths as scalable vectors as long as you want. They get rasterized in final export.
To have a wide palette you should learn both. Vector and raster graphics worlds are so different that a beginner gets easily confused. Learn by working through tutorials. Do not expect you can create something fine in the beginning. It's possible when the tools and are not the obstacle.
If you cannot afford both you can also start with low cost or free alternatives to learn the basics.