While it is very common, I find this sentence a bit ugly, and slightly incorrect. The noun adjective "viewer"—which I presume it must be—is ugly here, probably because it suggests that the entire noun phrase is a common term ("viewer discretion"), which I believe it wasn't until this formula was invented.
We could rephrase it as "we advise the viewer to use his discretion". We need "his" because the use of a noun adjective suggests possession. But "his / her / the King's discretion" is mostly said of people with some discretionary power, like a judge who has the right to make a decision based on his own judgement as opposed to on some external rule or power. Now, that is not what it is supposed to mean here; the phrase is somewhat misleading on that account.
What might have been intended was that the viewer should "use his own discretion" to decide whether he wanted to keep watching, which means his own ability to discern what the right course of action is in a given situation. But then we'd get "we advise the viewer to make a sensible decision": that is nearly a tautology. If this is the advice we get, then it is an out-of-control euphemism that hardly means anything. Moreover, the formula should then be "viewer's own discretion advised".
Alternatively, what is intended might be that the viewer should "use discretion": this means that he should be careful when considering his options, presumably because he will regret it if he doesn't give this decision the attention it deserves. This is the most reasonable explanation of the formula—but then it should have said "viewer is advised discretion": otherwise the viewer would be taken as the possessor of the discretion, which would take us back to "his discretion" as above. Moreover, "caution" would have been better then, because "discretion" usually means that you need to be careful in your behaviour with regard to its effect on others. In short, "viewer discretion", while not absurd or entirely impossible, is slightly off.
Using the passive voice in a notice like this—it may be a bit stiff, but it looks quite all right to me this way. It is clear that whoever is responsible for broadcasting the programme is advising the viewer.
I say it very much depends what Britons you are talking about. It is a dialectical style, found in a few places, but the most famous and that which tends to travel internationally is the East London, Cockney Rhyming Slang.
Now, most Londoners know a fair bit of it, growing up hearing it. But it depends what area of London you were brought up in, and more importantly what class you were. A lot of my friends use quite a bit of it, but not continuously like you see on films, it crops up as slang words, in the way that any other colloquialisms do.
Often, people do not know the etymology of the words they are using and do not know the rhyming part. Commonly used words such as barnet, boracic, china, cobblers, mickey, scarper and butchers are used frequently, but few know the rhyming component for them, they just use the words. Cobblers and mickey are more widely used than just London, not sure about the others.
As requested by Robusto, a brief set of rhyming expansions for the words I quoted above.
barnet -> barnet fair -> hair
Barnet Fair, was a regular horse trading fair held in the town of Barnet (my home town!), and well known by Londoners, as it was on the Great North Road, and was the major horse buying location for the capital. Further back it was the largest cattle fair in the country too.(Wikipedia)
boracic -> boracic lint -> skint
Boracic lint was an often used dressing for wounds, lint soaked in boracic acid. It rhymes with skint anyhow, which means pennyless, broke. Note, we don't pronounce it boracic, it is pronounced as brassic, possibly due to people thinking that there is a link to the word brass, considering the financial connotations. (brass is a slang term for money, particularly coinage - reinforced by the proverb "Where there's muck, there's brass.")(Wikipedia)
china -> china plate -> mate
This one needs little explanation, we had china plates coming in to the capital by the boat load, it rhymes nicely with mate, which is the most popular word for friend in most London areas. Though bruv gets used a lot these days in its place.
cobblers -> cobbler's awls -> balls
Balls being used in the slightly vulgar sense that bollocks is usually roped in for. To say that something is untrue, or rubbish in some respect, mainly refuting something said by another. The primary phrase I hear it used within is "What a load of old cobblers!"
mickey -> Mickey Bliss -> piss
Mickey Bliss was a bloke, supposedly from London, who no one knows anything about - he remains unidentified. Except in infamy for being the source of this rhyming slang couplet, quite a recent edition to the pantheon, 1930s I believe. Always used in the phrase "taking the mickey", although michael/mick can be substituted, from "taking the piss", which we recently had a discussion about. Note you can not use mickey for the act of urinating, the rhyming slang for that is jimmy (from Jimmy Riddle -> piddle).(Wikipedia)
scarper -> Scapa Flow -> go
The etymology on this one is a little unclear, it is known as rhyming slang from Scapa Flow (a very important natural harbour in the Orkney Islands) - but came additionally (and most likely originally) from the Italian immigrant population via their verb scappare which means to escape. The current meaning is to make a quick get away, we had a discussion on this word recently. It was used heavily amongst certain groups, notably it was taken up in the Polari language in the old phrase "scarper the letty", letty meaning bed/board/lodgings in Polari.
butchers -> butcher's hook -> look
We say, "take/have a butchers" to mean a quick look, synonymous with the word gander when used in that sense.
Hope that is of some interest to some of you, I do love etymology, particularly for obscure colloquialisms. :-)
Best Answer
Despite the fact that there exist many varieties of English, the vast majority of spoken media is in the General American or in the Received Pronunciation of British English. Written media is pretty dialect-free (again the great majority).
So your difficulty is probably understanding spoken language in a casual atmosphere. It could be that the people you hear come from a variety of backgrounds, but they will most likely be aspiring (if ESL speakers) to one of American or British pronunciation.
So the first step is to pick one of those. The general point is that no choice is wrong but making a choice now will solidify skills in one, to make learning the other one easier later.
As you say, movies (and TV) will teach you good pronunciation as far as accent. Yes, too casual speech (slang) might be confusing, and you don't want to learn profanity inappropriately (which is almost always). But one can often find online transcripts so that you can read along. Actors in TV and movies do tend to articulate well and in the standard. Some movies might use a 'bad' variety (mumble too much or be too casual/slangy/taboo), but most are not. Comedies might be hard because of word play, but then that would be a good learning lesson too.
News reports are good too because their pronunciation will be very articulate. Start with TV news rather than radio, because it will give semantic context with the pictures, which will help disambiguate unsure vocabulary.
A growing, very easy resource is youtube videos. A lot of it is amateur, meaning, not produced with expectations of high quality and experienced actors. It will be more casual and conversational, more likely to have dialect/pronunciation variations, but the real challenge is the natural slurring, mumbling, rushed lack of distinct articulation. I'd suggest sticking with the more professionally produced youtube videos before graduating to the real-life speech of the amateur (which is eventually what you really want to learn).
But frankly, don't leave something out if its easy to do. That is, listen, listen, listen. (Talking too is very important, in fact terribly important, the more you try to talk the more practice in your head you'll be doing and you'll be able to understand more that you hear)