Heading off north and east at first on river that runs into the Med (5)
I believe the answer is:
rhone
'that runs into the med' is the definition.
(the River Rhone runs into the Mediterranean Sea)
'heading off north and east at first on river' is the wordplay.
'heading off' suggests taking the first letters.
'and' is a charade indicator (letters next to each other).
'at first' suggests taking the first letters.
'on' is a charade indicator (letters next to each other).
'river' becomes 'rho' (I can't justify this - if you can you should believe this answer much more).
The initial letter of 'north' is 'n'.
The first letter of 'east' is 'e'.
'n' next to 'e' is 'ne'.
'ne' after 'rho' is 'RHONE'.
(Other definitions for rhone that I've seen before include "River running through the city of Arles" , "Major French river, rising in Switzerland" , "Camargue river" , "River running from the Swiss Alps to the Mediterranean" , "River running through France to the Med" .)