At its heart, English is almost two seperate languages, old German and Norman French. Vocabularies exist from both sources side-by-side. The German roots provide us with a language which is grammatically simple, Norman French with a multiple-syllabic vocabulary with often seems pompous. There was, of course, a social divide between the two language systems with the lower orders using the German base.
More correctly, the old German was actually the combination of several Germanic languages, especially those of the Saxons and the Angles - hence 'Anglo-Saxon'. As HH says, there was a social divide between those who spoke this language and the newer-comer, Norman French, which - in itself - is a Romance language based on Latin and Greek, as are all the languages of the Southern European nations.
However, it should also be remembered that long before the Angles and the Saxons arrived in Britain (early- to mid-5th century), there was already the Celtic language, varieties of which are now mostly spoken in Brittany, Cornwall, Galicia, Ireland, Isle of Man, Scotland and Wales, which vied with the incoming Latin and Greek of the Phoenician traders and Romans.
Even more historically, all these languages have a common ancestor - an early form of Sanskrit - which gave rise (according to some linguists) to the Proto Indo-European family of languages.
Since the arrival of the Normans, there have been tranches of additional vocabulary and even grammar as a result of borrowings from languages that the British have traded with and/or colonised. Modern English is an amalgamation of all of these factors and heritages, and we should not decry it's resulting complexity.
From a language learning p.o.v., it is possibly the 3rd hardest to learn as a non-native language, after Cantonese and Tamil (the language of Tamil Nadu in South India), though obviously, any such listing has to take one's own mother tongue into account.
As far as its spelling is concerned - there is no strong link between phonemes and graphemes. There have been attempts to re-order this. At one level, Noah Webster tried to do this with his dictionary - in contrast to Dr Johnson who tried to retain the eccentricities of spelling in his. George Bernard Shaw campaigned for the rationalisation of English spelling.
Every such attempt to rationalise the spelling of English has fallen foul of the very nature of the language - the variety of sources that its vocabulary has come from as mentioned above.
One of my favourite examples, which I've shared (here?) before, is that of a Nepalese nursing student questioning why a book on 'Pediatrics' was in the library's section on children's health. It transpired that she had had a voluntary English teacher from Britain (VSO, iirc) at school and she had been taught that anything starting with 'ped-' referred to
feet (pedestrian/pedal/pedicure/...). The book she was querying had been produced in the USA, where the same thing doesn't apply. Most of the books in the college library had come from Britain, so most child health books referred to 'Paediatrics'.
One of the promoters of a new orthography was Sir James Pitman, of the Pitman shorthand family. I don't do shorthand but I believe that it involves a fairly close association between phoneme and the shorthand graphemes.
I understand that the problem Pitman's system faced is that faced by any such system within English: you get a far larger number of homophones - words that sound the same, even if they don't mean the same - eg:
sew (repair or make clothing)/
sow (put seed into the ground)/
so (an adverb or conjunction). This actually makes it more difficult for learners.
