I think if you said any of those things to someone, they would understand what you mean.
I am going to [ . . . ] get
(you don't need the definite article here unless you have previously talked about groceries, what exactly you are going to get, or what exactly you mean by "groceries")
I don't find any of them very practical or natural. The use of "go" in all three options seems unnecessary after "going to".
You would normally hear
I am going to the grocery [store].
I am going to get groceries.
They are short and to the point. And they are quite natural and idiomatic. The following is correct, but I don't see why I would use this instead of the shorter versions when the context is particularly about going to the grocery:
I am going to go [out to] get groceries.
But note that the phrase "going to go to" is correct, idiomatic, and commonly used depending on context:
going to (v): intending to do something in the future, or being certain or expecting to happen in the future (Cambridge)
- Are you going to go to Claire’s party?
You could, however, say
You (or I) need to go get groceries; we are out of food.
I need to go out get some food.
I should go get groceries. ("out" is implied, no? The grocery store is outside, not inside your house)
I will go get groceries.
In this context, "go off" is the least preferred option. According to Cambridge dictionary, one meaning of "go off" is "leave":
"to leave a place and go somewhere else"
- She's gone off on holiday with Tony.
But this meaning does not apply to your case.