Let's look at each sentence individually.
I need to go out, get a few things from the store.
This kind of sentence is sometimes used in informal verbal communication, but it is not considered technically grammatical because get a few things from the store needs something like a conjunction to link it to the previous clause.
I need to go out, and get a few things from the store.
This sentence is perfectly fine, although I would say the comma is unnecessary (but that's a stylistic point).
I need to go out getting a few things from the store
This is the only one that is flat out wrong. There are two problems here: the forms of the two verbs should match, and there's no conjunction. To see why, you can think of the correct sentence I need to go out and get a few things as meaning I need to go out and I need to get a few things.
I need to go out. Get a few things from the store.
This is incorrect because the second sentence is a sentence fragment, or may be interpreted incorrectly as an imperative (as in Go get a few things from the store). Like the first example, this may be heard in informal verbal communication sometimes. The difference between the first example and this one is that the period indicates a longer pause.
Now let's consider the original sentence:
I need to go out to get a few things from the store.
This means something slightly different, in that it means I need to go out in order to get a few things from the store. However, this difference is extremely minor, as it is implied in the other sentences that the purpose of going out is to get things from the store. In this case, the purpose is made explicit.