A famous quote from the satirical political TV comedy Yes, Minister sums up the British press fairly well. Amusingly, one of the characters is named Jim Hacker.

Hacker: Don't tell me about the press. I know exactly who reads the papers: the Daily Mirror is read by people who think they run the country; The Guardian is read by people who think they ought to run the country; The Times is read by people who actually do run the country; the Daily Mail is read by the wives of the people who run the country; the Financial Times is read by people who own the country; The Morning Star is read by people who think the country ought to be run by another country; and The Daily Telegraph is read by people who think it is.

Sir Humphrey: Prime Minister, what about the people who read The Sun?

Bernard: Sun readers don't care who runs the country, as long as she's got big tits.

(The Mirror is a leftish tabloid, the Sun a rightist tabloid, both are strident and stupid. The Guardian is a lefty broadsheet, very 'liberal media' by American standards. The Times is center-right, the Daily Mail is middle-class outrage and celebrity gossip (and outrage about our culture of celebrity gossip). The Financial Times is like the WSJ without the axe-grinding editorials; the Morning Star is now defunct but used to be a mouthpiece for the USSR communist party. The Telegraph, sometimes referred to as the Torygraph for its unflinching support of the Tory party, is basically a serious newspaper for people who are convinced the country has gone to the dogs, global warming is nefarious plot, and so on. There's also the Independent, which didn't appear until some years later, which can best be summed up as 'worthy but boring.')

Broadsheet/quality press - fairly balanced in journalism. Editorially, while its more nuanced than left and right, this will give you a general idea of political leanings. Guardian is leftish, Telegraph is rightish, New Statesman is leftish.

The Daily Mail is the fox news of the UK. The Telegraph is not a whole lot better - it's actually less serious and dangerous and just ridiculous. The Guardian is pretty similar, just for the opposite political side.

The Telegraph is a serious, respectable newspaper; quite a way out on the right (at least by British standards), but with integrity, and open about its biases. The Guardian is more or less equal and opposite; again a serious, respectable paper, but one that caters to sandal-wearing vegan hippies. New Statesman is a very respectable magazine, probably analogous to The Economist (and with a similar right-wing but socially liberal stance). But yes, between them these are about as good as it gets, pretty much the opposite of "the fox news of europe".