I hold an Irish Passport and arrive in Georgia next month. I have read many comments about having Armenian or Azerbaijan stamp in the passport which might cause problems in Armenia or Azerbaijan. What would be my best route so I can enter each country?
You can visit both countries in any order as you wish. This is what many tourists do (when they visit a region, they want to see all of the countries in it) and both Armenian and Azerbaijani authorities are aware of that. As long as you don't enter Nagorno-Karabakh and underlying occupied territories, you won't have problems visiting Azerbaijan. From MFA of Azerbaijan:
The Ministry would like to remind that those who traveled to the occupied territories without prior permission of the Republic of Azerbaijan will be denied the entry to the Republic of Azerbaijan.
Although when you enter aforementioned territories your passport doesn't get stamped, still it's better not to do it, if you later want to visit Azerbaijan.