One wants to join another class and do something in the garden (6,7)
I believe the answer is:
social climber
'one wants to join another class' is the definition.
(someone who wants to join a superior social class)
'do something in the garden' is the wordplay.
'do' becomes 'social' (I've seen this before).
'something in the garden' becomes 'climber' (a climbing plant).
'social'+'climber'='SOCIAL CLIMBER'
'and' acts as a link.
(Other definitions for social climber that I've seen before include "status-seeker?" , "One aiming for higher status" , "Someone wanting higher status" , "Upstart" , "One seeking higher status" .)