How do I make 3d molecular graphics similar to those shown on Wikipedia?
You can generate the geometry for your molecule of choice by drawing it in Avogadro. More often than not, you want to refine the geometry through a force field calculation. 

Avogadro supports multiple ways of visualization, which all can be configured according to your preferences. A ball-and-stick model, which you seem to prefer, is included.

If you want to generate an animation from the input file, QuteMol might be one option.

The interactive visualization on the wikipedia page of benzene is generated using JSmol, which is a JavaScript implementation of Jmol.

If you are only interested in online tools, MolView could be an option.