EU Info point in every village, town, city, youth organisations within Europe.
As you may know in greek universities people orgnise themselves in political groups sort of the fraternities in usa colleges. These groups search and care for mainly for issues of political importance. Of course not everybody is into them and many people abstain from them. However if you get their attention I belive that they will have influence since every student votes for them directly or indirectly because every student is part of the students' society by law.
Through Structured dialogue to share ideas of young people on EU issues, or via variety of internship in EU´ institutions or NGOs dealing with EU issues to involve young people. Moreover, the central point how to involve more young people in EU issues should be a creating of a compulsory subject concerning the history, the structure and the function of the EU at schools all over Europe. By doing so, most young people get to know and better understand what the EU means and how it works.
Teach them to be proud on their heritage, they'll care for it.