Which one is correct to use ?
The bank manger was given a holiday and he resolved to go hitch-hiking.
The back manager was given a holiday and he resolved to go for hitch-hiking.
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To hitch-hike is a verb.
So people go hitch-hiking in the same way that they go swimming or climbing or similar, where the verb appears as a gerund.
On the other hand, one goes for a swim or a climb.
That's to say, where nouns are concerned, one goes for something. Where gerunds are concerned, one goes sailing/singing/hitch-hiking or whatever.