I am web designer and I have been working with Photoshop. Sometimes I feel like I need to move to illustrator when need of graphic design came. Do I need to do that and how much?
Having knowledge about how to use the standard 3 programs for graphic design is a definite advantage. The 3 being: Photoshop, Illustrator and InDesign. There are obviously alternatives to each one which are more or less viable, the base skills are still very valuable.
Photoshop is for editing images, Illustrator for editing vector images and InDesign for layout. Photoshop is one of those weird do it all kind of programs, but Illustrator will do its thing a lot better.
As a web designer, the value of knowing InDesign is more debatable as it is primarily for print and not anywhere close to great for web work.
What you could look into is Adobe XD or Sketch.