is the phrase grammatically correct? or should I say "The holidays are good times..." ? can we use "a" after "are" ?
The sentence is grammatical, and there’s no reason why are cannot be followed by a. Whether you write ‘The holidays are a good time . . .’ or ‘The holidays are good times . . .’ depends on what you’re trying to say. If you’re referring to Christmas, then the first is probably what you want. The second would be unusual. If you wanted to talk about holidays in general, then it would be ‘Holidays are good times . . .’
The holidays are good times to spend with the family (in the body of your question).
The holidays are a good time to spend with the family.
It depends on the plurality of
Singular, precede with
a, otherwise drop it.
Deciding which to use, would really depend upon the context of the sentence.