Перевод - Греческий-Немецкий - Εμείς πηγαίνουμε εκδρομή.

While this is correct, I also think that perhaps the French word "excursion" might possibly translate better to "Ausflug" in German. So I would suggest something like "Wir machen einen Ausflug"

I can only judge by the French translation. ' partir en excursion' is a more specific way of travelling than 'auf Reise gehen'. Therefore I think that 'Wir gehen auf Exkursion' or 'Wir machen einen Ausflug' would be more accurate.

I agree with bgl88.

I would say "Ausflug" instead of Reise.

I go along with the other suggestions (Ausflug or Exkursion).