Quick Definition of Vector Graphics:
Vector graphics are graphics that computers render by using mathematical formulas. This allows the image to be scaled up many times without distortion, unlike something in MS Paint which uses raster graphics (graphics made up of pixels).
Vector graphics are primarily used for logos, icons on your computer, and wallpapers due to their scalability.
(This is written for active learners; a step-by-step how to would cripple creativity.)
Simply play with the software and become familiar with the different tools for 15-20 minutes. Put 5 different shapes on the paper and a curve. Add different color fills to the shapes, and a gradient fill to one.
Redesign an icon from your desktop: Choose an icon from your desktop, and make a vector drawing of it, or of something else you would like to use as the icon. For example: headphones for your music player. Keep the detail low as it will be exported in icon size as a raster graphic.
Create a vector from an image: Choose a simple image, such as a flower or some object, and create a vector image of it (probably using curves). Don't use color or shadows yet. The primary goal of the day is to gain experience creating complex shapes with the program you use.
Add color to the objects you created in Day 3's assignment, and use a new object with darker colors to replicate shadows. Take note that vector drawings are usually just this, collections of objects with slightly different colors that look like something.
Practice, practice, practice. Just keep practicing! With enough practice, you should be able to start designing things like wallpapers with your vector graphics.