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I'm somewhat confused about the difference between these two constructions. I can't get when I should use a gerund and when an infinitive.
I believe it's very easy for a native speaker to tell the difference if it exists, of course [Emotion: party]

I'd also want to learn the British equivalent to the phrasal verb 'try out for'. For example, what verb would a British native speaker use in the following sentence:
Rita's trying out for the school play again. I have failed to find any verb which could replace 'try out for'. Hope you help me.
Comments  
I don't know what a more colloquial phrase would be in Britian, but how about "audition"? I'll have to give more thought to the other question.
"go out doing something" is like "go out while doing something", "go out in a certain manner" described by the specific 'something' in the sentence.
"go out to do something" is like "go out in order to do something", "go out for the purpose of doing something".

Note that in the first construction "go out" can be a literal going out or a figurative (idiomatic) one.

Literally, "The detective was so angry that he went out muttering to himself": He left the premisses; he was muttering to himself at the time. "to go out" = "to leave", "to exit"

Figuratively, "The team won the final game of the season. It was quite a thrill (for them) to go out winning": They finished/ended the season as winners. "to go out" = "to finish (a season, a career, one's life)".

Also, again figuratively, "The poor guy finally died of cancer, but at least he didn't go out whining about the injustice of life": He didn't die (bitter and) whining. "to go out" = "to die".

The second construction is normally literal: "Let's go out to get some fresh air". "They've often gone out to get ice cream at midnight".

CJ

P.S. I'm not familiar with any British expression for "try out for".
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Hello

Actually "try (out) for" is used in BrE. OED says "try for : attempt to obtain/find". (EX) She (= Jennifer Lopez) tried for the role of Evita but was beaten by Madonna.[ [url="http://www.guardian.co.uk/2000/article/0,2763,196741,00.html "] Guardian 2000[/url]]

You would say 'go (out) to buy a computer' rather than 'go (out) buying a computer' and 'go (out) to attend the meeting' rather than 'go (out) attending the meeting'. But you would say 'go (out) shopping', 'go (out) walking' rather than 'go (out) to shop' or 'go (out) to walk. The form of 'go (out) to V' is a grammatical usage of infinitives 'to V' [purpose adverbials]. The forms of 'go (out) V-ing', however, are quite idiomatic collocations stemmed from Old English where they said 'go V-ing' (here was a preposition meaning either 'on' or 'into'). We English beginners cannot know which verb preferably takes the idiomatic form 'go (out) V-ing, and so we have to learn them one by one consulting with dictionaries.

paco
I love this forum because it makes me really think about the speech patterns I use automatically. As paco says, some verbs are more likely to be used with one construction tht another - I would never say "let's go out buying a computer," but I would say "let's go out shopping." And I think paco is correct in saying that there's no rule for which verbs work which way .
In the case of verbs that can be used in either form, I would say there is a very subtle distiction - "go out shopping" stresses the process, and "go out to shop" stresses the purpose or the result. "I'm going out shopping" -- I'm going to spend the afternoon looking around the stores. Maybe I don't even need anything in particular, I just enjoy the activity. "I have to go out to shop" - I just realized I don't have everything I need for dinner. You could even say "I'm going out to walk" if you had a goal of walking 30 minutes for exercise every day, whereas "I'm going out walking" would just be for the enjoyment of it. Does that make any sense? As I said, it's a subtle distinction.
Hello Khoff

Thank you for the nice comment. Yes I can feel somehow there would be some distinctive tendendency/rule in your choosing the form of 'go out V-ing. I feel the choice seems to do with your traditional ways of enjoying life. But it's quite hard for us to catch the rule exactly.

paco
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'Audition' would I think be the usual BrE phrase.

Perhaps the paper felt 'audition' was too humble for Jennifer Lopez or Madonna. (Or maybe theatrical folk here use 'try out for' too.)

{wavy lines and music}

'Now then, you quaint little British-speaking person...I think I'll just try out this role of yours and see how it fits...'
'Certainly, Miss Lopez...'

{Hours pass. Bottled water is flown in. Zips and curses are heard.}

'...Okay, big boy. What do you think?'
'Well, Miss Lopez...'
'You're saying you don't like it, right?'
'I would never go so far as to say that, Miss Lopez. But...'
'Butt? Did I hear you say "butt"?'
'By no means, Miss Lopez! I wouldn't dream of saying "butt". Now, as I was about to say...'
'"A$s?" You were about to say "a$s"? Do you have any idea how much this thing is insured for? And you have the nerve to say "a$s"?'
'Please, Miss Lopez!'
'Here! Take your stinkin' role! And you know where you can shove it, Mr tea-sipping, denture-wearing, limp-wristed, stiff-upper-lip-biting resident of a country the size of Kylie's —'
'Well, if it comes to that, Miss Lopez, if you don't mind my saying so, your bum did look remarkably big in it. '
'You leave Puffy outta this! Anyway: ex-bum. You Brits are so last decade...Okay, it's all yours, Madge. Now: take me to my people. They await me with open arms. Am I not theirs? And they mine? Do they not know me as Evie on the Block? For my greatest fear in life is to be forgotten. Forgotten by posteriority...'

{***}
*quite a large one