Going round New Jersey, first support dog (7)
I believe the answer is:
basenji
'dog' is the definition.
(I know that basenji is a type of dog)
'going round new jersey first support' is the wordplay.
'going round' becomes 'base' (I can't explain this - if you can you should give a lot more credence to this answer).
'new jersey' becomes 'nj' (abbreviation).
'first' becomes 'i' (I in Roman numerals eg Elizabeth I).
'support' says to put letters next to each other (in a down clue, some letters hold others up).
'base'+'nj'+'i'='BASENJI'
Can you help me to learn more?
(Other definitions for basenji that I've seen before include "Barkless dog" , "Smallish hunting dog" , "Dog unable to bark" , "Breed of African hunting dog that rarely barks" , "Small hunting dog" .)