I need a change. I need going/to go away for a while.
Would you please show me which one you use and why?
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I would say:
I need to go away for a while.
The situation with need is somewhat complicated. You can put a gerund after need, but it suggests that the subject of need will be the object of the gerund. For example:
The house needs painting.
means the same as:
The house needs to be painted.
Adding to the confusion, if the verb after need means the same thing actively and passively, then you can use either the gerund or the infinitive:
The cows need feeding = The cows need to be fed = The cows need to feed.
because one sense of to feed means the same as to eat. This really isn't an exception, of course, so it doesn’t need memorizing. Or in other words, you don’t need to memorize it.
I don't think there is any general rule that tells which verbs should be followed by an infinitive and which verbs should be followed by a gerund. You just have to learn how each verb is used, one at a time.
This question on ELU has some more information.
You could say “I need going away for a while” to express strong emphasis. This wording is unusual, and suggests to a listener to hear “going away for a while” as if it were something done to you, like a medical treatment. This is tricky use of English, though, not something to start with, and not something you commonly hear. You should master the ordinary need to and need verbing usages first. But this illustrates the way you can use the grammar to make a listener understand a phrase differently than usual, if the listener can find an interpretation that makes grammatical sense.
Whatever you do, don't read this question about a nonstandard usage of need unless you want to sound like you’re from Pennsylvania or southern Scotland.