I think there is a Thread that covers everyone's opinion on how to hard boil eggs. LOL, you'll get an
ear eyefull.
Rule #1. Do not use farm
fresh eggs. (Universal rule, they are impossible to shell)*
after that a multitude of rules apply depending on who's telling you.
Starting at the fat end where the air pocket is. Crack the shell a lot, then I use a teaspoon to slide between the egg and shell. Works for me!
*I raised my own chickens for eggs at one time. I always kept a doz. at the back of the fridge to age about 2 weeks, just for hard boiled. If I didn't have them I went and
bought 
some to boil.
And yes, uncooked eggs, in their shells, keep longer than boiled. Shells are porous. When laid, the chicken deposits a coating on them to protect from bacteria. Boiling them removes this 'bloom', as it is commonly called, making the shells very susceptible to invasion.
In North America, store bought eggs are normally 'washed'. This makes it important to keep them in the fridge. In Europe they are not commonly washed and therefore often just kept on the counter.
Here endeth the quickie lesson.
Edit: more FYI
To help centre the yolk when boiled, if you can remember ( I never do)
put the carton on its side the night before you boil them. It is not 100% but works enough to be impressive... and not get those yolks so close to the edge it makes it near impossible to get them out without tearing the whites. enjoy!