Q:

Is there any difference in meaning between "go to do something", "go off to do something", and "go out to do something"?

Is there any difference in meaning between go to do something, go off to do something, and go out to do something? For example:

I am going to go to get the groceries.
I am going to go off to get the groceries.
I am going to go out to get the groceries.

A:

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. 

A:

To answer your question:

The "off" emphasizes leaving or being away longer than expected.
The "out" suggests exiting a building.

In conversation, most listeners wouldn't pay much attention to those differences or attribute much meaning to them. They would get lost in the "semantic noise" of everyday speech. 
If someone you knew used one of those phrases, you wouldn't be able to argue that their choice of words meant anything specific. But, if you were writing dialog in a play, you could use these subtle differences to imply something about the character's intentions.
About those examples... 
You don't need to repeat the verb, to go. It is simpler to say:

I am going to get the groceries.
I am going off to get the groceries.
I am going out to get the groceries.

You only say "the" before "groceries" if you mean specific groceries, like the ones in your car parked outside. 
If you're talking about going shopping for groceries, you don't use an article. This is because "groceries" is a noncount noun, something indefinite in quantity and not customarily counted. 
If you're going shopping, you would probably just say:

I am going to get groceries.

