How can one wish for different occasions? The safest trick we can give you is to use the adjective «καλός» (kalós) in the correct form to wish for different things. You know already some words: καλημέρα (kaliméra), καλησπέρα (kalispéra), καληνύχτα (kaliníhta), καλά Χριστούγεννα (kalá hristúghena), καλή χρονιά (kalí hroniá). You only need to know the gender of the noun!
For example, it is customary in Greece to wish “happy new month” on the first day of the month, implying health and happiness. For that, we say in Greek «καλό μήνα» (kaló mína) and it can be accompanied with «με υγεία» (me ighía), that is “with health”. It is also popular to wish «καλή εβδομάδα» (kalí evthomátha), that is “happy new week”, on Monday at work, at school etc.
These are called «ευχές» (efxiés), meaning “wishes”, and they are commonly used in our daily conversations. Another example is «καλή αρχή» (kalí arhí) or «καλό ξεκίνημα» (kaló xekínima), that is “good start” or “good beginning”, when someone starts a new job, a new hobby etc.
One might also wish «καλή δύναμη» (kalí thínami) which means literally “good strength” but indicating “be strong” when in a difficult situation.
You might have heard the «καλή όρεξη» (kalí órexi) that is “bon appetite” or «καλό ταξίδι» (kaló taxíthi) that is “have a nice trip”.
Bonus: we have an idiom that says «καλά κρασιά» (kalá krasiá), literally meaning “good wines” but actually we say that for something we find unlikely to happen!
Feel free to experiment yourselves with the adjective «καλός» (kalós) and wishes!