I think if you said any of those things to someone, they would understand what you mean.
I am going to [ . . . ] get the groceries.
(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.