Recommend me some quintessential British fiction

Rivers of London by Ben Aaronovitch is modern and has very dry, British humour. It's set in London and is about a policeman. It's a bit 'out there' though. I loved it.

The books of Jane Gardam (Old Filth, The Man in the Wooden Hat) and Patrick Gale.

Possibly the English equivalents of Anne Tyler? Sort of territory.

Wizardofos I am obsessed with London Belongs to Me. Have also just started Bond Street Story which is out of print but got it cheap on Amazon.

OP, not quite sure what sort of thing you like but I love:

Vanity Fair (several mentions and yes, it's oddly modern)
Antonia White's Frost in May series
Diary of a Nobody - George and Weedon Grossmith
Diary of a Provinical Lady - EM Delafield

As for more modern stuff, Martin Amis has written some good books despite being fairly annoying. London Fields I think is his best and most British. I really loved Hilary Mantel's Wolf Hall and Notes on a Scandal by Zoe Heller.

The Lady in the Van by Alan Bennett. He has written lots of other great things too but my brain is too fuzzy to thing of them.

contemporary-ish Scottish crime fiction:
Ian Rankin Rebus series, the earlier in the series the better general
Denise Mina Garnethll trilogy
Peter May The Blackhouse or the Lewis Man (set in Scottish islands)

Alana - I know what you mean but I think you have to include some Irish stuff if you want to know about GB generally. In fact I would recommend that the original poster explores Irish literature in general - a goldmine!

Twenty Thousand Streets Under the Sky
Slaves of Solitude
Hangover Square

All by Patrick Hamilton

London Belongs to Me by Norman Collins

simon brett
dick francis
are my favourite crime writers

Chubbleigh I could add lots of Irish stuff but then it's not really British is it? Not the stuff I'm thing of which is all Ireland not Ulster.

Thackeray generally, not just Vanity Fair. For example, Barry Lyndon and Henry Esmond are very good. Funny too.

Graham Greene and Le Carre more recently(ish).

These lists need some more Scottish/Welsh/Irish content and a some titles that are a bit less posh. Really random thoughts:

How Green was my Valley
Kes
Trainspotting

Also some wartime titles, Regeneration trilogy - Pat Barker, maybe

I read a lot of Australian books when I lived there and enjoyed reading autobiographies in between the classics, mostly of of the not at all famous, just those with a good tale to tell. Example, Call the Midwife, Jennifer Worth, the books are a good read and very detailed about day to day life.

Oh and I Capture the Castle obviously but it MUST have been mentioned dozens of times already!

And if you want to go beyond English, please to kindly read John Burnside's memoirs, Waking Up In Toytown and A Lie About My Father (JB is THE Scottish writer, IMHO!)

Another vote here for pretty much everything reissued by Persephone Books

Dorothy Whipple 'They Were Sisters' is astonishing. One of the earliest honest, painful treatments of domestic violence. And a wonderful story of love and hurt between sisters.

Most of what I was going to suggest has already been covered, Jonathan Coe's books for example. Muriel Spark, not just Miss Jean Brodie great though that is, but also some of her novels set in 50s/60s London, Loitering with Intent is great and I'm trying to remember the other one I loved last year - The Ballad of Peckham Rye is worth a read too but there was one I loved more.

I don't think anyone's mentioned Winifred Holtby's South Riding, a brilliant novel about 1930s local government in rural and small town Yorkshire which is regarded as the author's masterpiece. I also really enjoyed Anderby Wold, her first novel. All her books have been published in Virago Modern Classics, The Crowded Street is currently a Persephone, but they're all out of copyright now as she died more than 70 years ago and there are currently two editions of South Riding on sale here.

You could do worse than read crime fiction - a mixture of "Golden Age" and modern day stuff.

Fishpond what would you recommend me as quintessential American fiction.
Here in NZ the online booksellers are called Fishpond - is there a link?

The Go-Between is a goodun. It's available on US Amazon here

Would you read the Agatha Christie books?

I've just thought of Elizabeth Gaskell - 19th century social novels set in the north of England and much better than that makes it sound. My favourite is North and South, closely followed by Mary Barton. A lot of people like Cranford, but I don't.

I second The 39 steps.
Other classic spy/thriller: The Riddle of the Sands by Erskine Childers, Rogue Male by Geoffrey Household. King Solomon's Mines by Rider Haggard for colonial adventuring.

And I also find India Knight very funny, but I know that some loathe her with a passion.

Ooh, have realised that most of my list have already been covered, but forgot to add The Enchanted April by Elizabeth von Arnim - a lovely, joyful book about lonely, miserable women cheering up a lot on holiday in Italy.

Neville Shute; despite often being abroad I always feel is very 'british'

I agree with loads of the recommendations here, particularly Waugh, Trollope and I Capture the Castle.

Just wanted to say on availability, if it works the same in reverse, you can use your Amazon ID & password to buy on any of the Amazon sites worldwide - I am in the UK now, but used to buy from Amazon.com, .co.uk and .co.jp when I was living in Japan (would check availability & postage on all of them to get best deal), and have bought from .com while in the UK (US books & DVDs not yet available here). Obviously you will pay more postage if you buy from the UK, but it's not always as bad as you think.