Very early start by Mediterranean city, a major tourist destination (6)
I believe the answer is:
venice
'a major tourist destination' is the definition.
(popular tourist destination)
'very early start by mediterranean city' is the wordplay.
'start' suggests taking the first letters.
'by' is a charade indicator (letters next to each other).
'mediterranean city' becomes 'nice' (I've seen this in other clues).
The first letters of 'very early' is 've'.
've'+'nice'='VENICE'
(Other definitions for venice that I've seen before include "See 1" , "Wonderful Italian water-city" , "Car-free city" , "major tourist attraction" , "It used to be known as the Queen of the Adriatic" .)