"My family is pretty good recently."
is the meaning of 不错 really "pretty good"? I thought that meaning of 不错 is "not bad" and "pretty (very) good" is 很好....
No. As a native speaker, 是 is seldom used between noun and adverb. 是 is more often used like:
between noun and noun: 他是特朗普。 <-> He is Trump.
between noun and adjective, mostly needs 的 at the end of the sentence: 我是英俊的 <-> I am handsome.
Is saying '我家人‘ instead of '我的家人’ incorrect? As a bit of a scruffy bilingual I'm trying to neaten my grammar up, and I know that sometimes in colloquial speech '的‘ is sometimes omitted, but is it grammatically wrong?
The English should be "My family has been pretty good recently". "Is" is present tense, while "recently" is past tense, so they don't agree.
很好 does not necessarily mean very good, as 很 is generally used and therefore isn't particularly strong as an adverb. So I see no reason why saying 很好 as opposed to 不错 is an issue. Also, would 家 not suffice as a translation to family? Perhaps it isn't as common/clear as 家人, but the latter refers specifically more to the family members as opposed to the family as a whole, no?