Graphic design and illustration are related but not necessarily the same thing. There are many excellent illustrators who are not working in graphic design. Graphic design can include anything from posters and album art to advertising, and into web design and so on.
Photoshop is only one tool that graphic designers and illustrators may use. Illustrator is also used, and for many aspects of graphic design is the better tool. For example, if you are producing a logo to be used on a variety of materials, from business cards to letterhead, to brochures, to billboard sized ads, you would want to use Illustrator because it is a vector program (Photoshop is a raster program), and vectors can be resized without loss of quality, while raster images cannot.
Many digital artists (graphic designers, etc.) use a graphic tablet because it gives you more flexibility and control over what you do.
Many digital artists also start out with "traditional" materials: pen or pencil and paper, watercolors, etc. They may work up a good quality rough in these materials, then scan them in using a scanner to digitize the image, and then go on to work on the digital version using Photoshop, Illustrator, etc. Many, if not most, good artists will work up a series of thumbnail sketches first, to play with concepts and ideas, before they go on to more finished work. There can be many steps from original idea to final image.
Hopefully the good art you are seeing does NOT involve copying the pictures from the Internet. For one thing, if someone does this and represents that image as their own, they are breaching copyright.
I recently got my BFA in Graphic Design.