What is the right way to say “congratulations” in Greek?

Sofia Mouratidis is right. She’s also right in the formal synonyms, and in one of the informal synonyms.

I’ll add a second informal synonym: συγχαρίκια. Amusingly (to me anyway), the original meaning of συ(γ)χαρίκια is “congratulatory gift”. When you brought someone good news, they were expected to reward you with a synkharikin. In fact, before telling the good news, it became a thing to tease the lucky person with τι θα μου δώσεις για συγχαρίκι, “what’ll you give me as a gift? (for me bringing you the good news)”. As the custom died out, people retained that synkharikin had something to do with good news, and just used the plural as “congratulations!”

But I gotta say, the true colloquial equivalent of “congratulations!” is a simple μπράβο, “bravo!” In Greek, it’s more like “well done! good for you!”