Family pet's first flower (6)
I believe the answer is:
catkin
'flower' is the definition.
(catkin is a kind of flower)
'family pet's first' is the wordplay.
'family' becomes 'kin' ('kin' can be a synonym of 'family').
'pet' becomes 'cat' (I have seen 'Common pet ' mean 'cat' so perhaps 'pet' could also mean 'cat').
'first' says to put letters next to each other.
'kin' put after 'cat' is 'CATKIN'.
(Other definitions for catkin that I've seen before include "Flowering shoot of willow etc" , "Ament -- on the hazel tree" , "Flowering spike hanging from willow and hazel" , "Feature of willow/hazel" , "descending from tree?" .)