It's best to appear in public well-dressed (8)
I believe the answer is:
outsmart
'it's best' is the definition.
(outsmarting is a kind of besting)
'public well-dressed' is the wordplay.
'public' becomes 'out' (out can mean released to the public).
'well dressed' becomes 'smart' (I've seen this before).
'out'+'smart'='OUTSMART'
'to appear in' acts as a link.
(Other definitions for outsmart that I've seen before include "Outwit" , "Fox; deceive" , "Outmanoeuvre cleverly" , "Cunningly get the better of" , "Defeat by skillful maneuvering" .)