Restaurant's first to open branch and last to go (6)
I believe the answer is:
bistro
'restaurant's' is the definition.
(bistro is a kind of restaurant)
'first to open branch and last to go' is the wordplay.
'first' becomes 'ist' (**).
'to open' indicates putting letters inside (some letters create an opening in a word).
'branch' becomes 'br'.
'and' says to put letters next to each other.
'last to' indicates one should take the final letters.
The last letter of 'go' is 'o'.
'ist' going inside 'br' is 'bistr'.
'bistr'+'o'='BISTRO'
(Other definitions for bistro that I've seen before include "A small informal restaurant from France" , "Small inexpensive eatery" , "Intimate eatery" , "Cafe/restaurant" , "Coming out of restaurant" .)