As I write this post, it's Christmastime in the year 2020. Time for holiday-themed emails, greeting cards, and gifts. The Word Doctor is getting involved by providing this valuable public service: holiday composition etiquette!
Let's begin with possessive vs. plural.
Seasons greetings - nope...not plural!
Season's greetings - greetings of the season, possessive.
Likewise, Happy holiday's - incorrect! the apostrophe indicates possessive.
Happy holidays - correct! Plural holidays.
Merry Christmas from the Smith's - incorrect, unless you include the word "house."
Merry Christmas from the Smiths - correct...ALL the Smiths!
Happy New Years - incorrect in several ways. Years is plural, so unless you're sending greetings intended to last for several years, that's just wrong. We're talking about just this coming new year!
And likewise, Happy New Year's is wrong unless you include the word "day." Happy new year's day is correct.
And as for capitalization, for most greetings standard sentence capitalization applies - only the first word.
Happy new year!
The exception is Christmas - Merry Christmas, because the word Christmas is always capitalized.
Now you're all set to go forth and dazzle associates and friends with your grammar acumen!