Hi Panos
You are able to Tavel to the Republic of Ireland with your EU passport.
You can find all the official information
here.
As a citizen of a European Economic Area (EEA) member state (27 countries of the European Union, together with Iceland, Norway and Liechtenstein) and many other countries including Great Britain, the USA, Australia, Canada and New Zealand, you don't a visa to visit the Republic of Northern Ireland.
More information, about customs and borders can also be found on the Ireland.com page
here.
Please let me know if you have any other questions regarding your visit.
And if you are looking for tips on how to spend your time on the Emerald Isle, I'm happy to send you some suggestion.
Happy Planning!
Megan.