Jewish clergy wife here.
There are a LOT of topics and questions here! I am sure we Jewish Brimstoners are all happy to answer any questions as they come up or occur to others -- I'm sure everyone would prefer that to a long essay trying to cover too many things (there are books available for anyone who wants that much information at once).
To respond to the original question (and a lovely question it is), don't worry about trying to learn Hebrew or Yiddish greetings for each holiday. This time of year, "Happy new year" will do fine, even around Yom Kippur (another poster was correct that you don't wish someone a "happy Yom Kippur" or "Have a nice Yom Kippur"), because Rosh haShana and Yom Kippur go together, referred to as "the high holidays." So even if it's closer to Yom Kippur, "Happy new year" or "Have a sweet new year" is an appropriate seasonal choice that will be appreciated. That's all you have to do for your friends.
The other thing that we really appreciate regarding holidays is people being aware of the calendar and trying to avoid scheduling community/school/etc. events and meetings on holidays -- it can put us and our children in awkward or impossible situations. Look at any basic calendar and it will have the high holidays at least, and possibly the big three festivals, Succot, Passover, and Shavuot. (Be careful googling for a calendar of Jewish holidays -- a few will have an astonishing number of obscure fast days and minor things that you definitely don't need to worry about.) It's also polite to avoid calling about business matters, going door to door, etc. on Jewish holidays (don't call at all on holidays or the Sabbath, Friday night through Saturday, if they are observant).
I hope that helps.
Happy new year!