The BBC isn't really neutral. I've had two encounters with them personally (Iraq war protests and long ago when I worked with the NHS during the initial Connecting for Health program). In both cases I have seen BBC journalists knowingly misrepresent things. Also, when you move away from pure news and go to their other shows, you find starker examples of bias. You only have to watch a little of Andrew Neil's shows to witness his Bill O'Reilly attitude to the truth. They do however, value the appearance of neutrality very highly.
In general, there are two approaches to getting real news. The first is to go to
financial sources of news. My gold standard for reporting is the private forecasting agency
Stratfor. Unlike the general mass of news audience who approach news as a method of either entertainment or validation of their existing views (and a supply of factoids to support those views in argument), customers of financial and investment news sources require solid analysis and reporting that reflects reality because it directly impacts their business. An investor doesn't want to read that the people in the Crimea are good or bad, they want to know how organised they are, how much the USA is backing them and whether they'll end up as part of Russia or not. I know this is a contentious example so I'll try and keep it objective and to the point. Most of the people in the Crimea are ethnically Russian, want to be part of Russia and regard the overthrow of their president during the "Orange Revolution" as illegitimate. The Daily Mail wont lead with that! They wont say things like "the US-backed revolution" about it, but Stratfor will and does. That's not because Statfor isn't a US-based company without a predominantly US market - they're very much both these things. It's because their customers aren't interested in being supplied moral talking points. And quite frankly, as an audience made up of hoary old Real Politick die-hards, they regard moral justifications as a glob of sugar for the masses anyway. Genuinely, Stratfor is the least biased and well-informed news source I have yet found. I hope the particular example I picked doesn't detract too much from that by triggering reactions - it's simply the first that came to mind as something that gets a lot of emotive spin in the mainstream media but which isn't included in their coverage. If anyone thinks that's an example of bias, I assure they don't present Russia's assumption of the Crimea as anything other than a powerplay either. Their goal is to guide investors as to what is actually happening. Their analysis on Trump's policies and what it means internationally has been very good. Most media treat Trump as if he can single-handedly choose whatever he wants. They've been exploring the factors that constrain him.
Of course, Stratfor is a paid service (and not cheap). It also has a very limited remit which is geopolitics. However, the principle holds just about in other sources - the Financial Times is solid. Again for the same reason that people actually USE the information therein and will go elsewhere if it fails them. I have to add a caveat about The Economist, though. It has a very good level of both coverage and depth and I read it semi-regularly, finding it very good. However it does have a very notable bias that trips them up not infrequently. I recall them confidently asserting that Hollande would not be elected in France because he was "too socialist". They had clearly never met the French. So The Economist is good but very much immersed in their own economic theories. They don't have a bias to any political party particularly, but they do have a bias towards their own economic theories - all the more dangerous because they seem unaware of their bias. I'll give the Daily Mail reporters one thing - I'm pretty sure they all know what they do for a living and don't regard themselves as enlightened ones!
I'm also (and here come the boos) going to give a shout out to
Russia Today. They have good coverage of subjects. They don't have a huge amount of depth. They are not, despite, accusations, a simple propaganda mouthpiece for the Kremlin and they frequently cover things that you wont find in the mainstream media. Even those here who don't like them will concede the following principle: If you want to know what your enemies are doing wrong, read your own media. If you want to know what you're doing wrong, read your enemy's.
And that final point illustrates neatly that no single source will provide you with an accurate view. Like diet, you need a balanced intake. You also need the willingness (hard to come by) to put aside what you think you know and accept you can be wrong some times.
I'll also give a shout out to The Guardian because despite terrible bias and some horrendous opinion pieces (including one arguing that the judge in the Keysha case should have found the company guilty despite the lack of evidence on the grounds that the accusation was so serious - yes, really), they are one of the few British papers that actually still employs real journalists. It wasn't the BBC that Dr. Kelly went to with his information about Iraqi WMD (and ultimately paid for his life with). It wasn't the FT that Edward Snowden supplied encrypted files to or whose journalist was detained by the police by.
A really good exercise to do, is to swap around your news sources for a period of time, and really read them and question them. I would suggest reading Russia Today for a week every day, and the Telegraph the next week. Read the Financial Times for a week and the Guardian for another week. All are good quality news sources coming from different angles. (And no, usual suspects, I'm not going to retract RT from that list - if the OP is really wanting to strip bias, they should include that not skip it because of what they have heard about it).
Anyway, I hope that helped or was at least interesting. I typically read The Economist, Stratfor (paid), The Telegraph, The Guardian and Russia Today and I alterante between those roughly once every few months.