How will I start this proof? (A set being open) Show that:
$\left\{{p :  |p-p_o|>r}\right\}$ is open for any $p_o$ and any $r\geq0$.
How will I start this proof? What should I need to show i.e should a delta neighborhood needed? If so, how should I use it?

Fix $r$ and $p_0$
Show that every point in your set is an inner point by finding a neighborhood that is contained in your point. 
To find such a neighborhood remember that a metric satisfies the triangle inequality. Rearrange it to get a sufficient condition such that points in the neighborhood remain in your set.
I hope I haven't given away too much.  
