A literal translation of the expression 'Family First' into Italian would be 'Famiglia Prima', however this word for word approach would probably draw a few frowns of confusion from native speakers. A more natural translation would be 'La famiglia prima di tutto' or 'The family before all else'.
If you're looking for a short phrase that symbolizes strong allegiance to the family unit, perhaps 'La Famiglia per sempre' would be an appropriate one, translating as 'The Family forever'.
These kind of phrases could be used for all sorts of things, with family Christmas cards, jewelry engraving and personalized tattoos being a few great ideas.