It would depend on what you are trying to say.
In your first example
Family is important to me.
would be understood to mean a family unit/structure has importance.
If you said
The family is going on holiday.
it would be understood to mean they are all going together to the same place, whereas
The family are going on holiday.
may be understood to mean the same or to mean the family is holidaying at the same time, but possibly in different places (not together), further context would be necessary to resolve the ambiguity.
However, your sentence
Family are important to me.
would not exactly be incorrect but would be understood to mean that each individual in the family are important.