Dandy follows celebrity to begin journey (5,3)
I believe the answer is:
start off
'begin journey' is the definition.
'dandy follows celebrity' is the wordplay.
'dandy' becomes 'toff' (I've seen this in another clue).
'follows' means one lot of letters go next to another.
'celebrity' becomes 'star' (I've seen this before).
'toff' after 'star' is 'START-OFF'.
'to' is the link.
(Other definitions for start off that I've seen before include "Initiate" , "Make first move" , "get to work" , "Begin" .)