Read some first-rate literature
The Roman poet Catullus, for example, who wrote lovelorn poems to his mistress (who is alleged to have poisoned her first husband). Or his contemporary Cicero, the great barrister and politician, eventually murdered on Mark Antony's orders. His speeches show unrivalled mastery of persuasion; his own dreadful poetry has, mercifully, perished with him. Or Thucydides, who declared in his preface that his work was 'a possession for eternity'. He was right: his history of the Peloponnesian War is still read 2,500 years later.
Yes, you can read their works in translation, but that is a pale shadow of the original. Imagine Hamlet's speech 'To be or not to be' translated into French. Of course a French speaker could then understand it, but the lilting rhythms and stark simplicity of Hamlet's expression would be irretrievably lost.
Understand European history and culture
The Romans invaded Britain in AD 43 and built roads and towns that are still in use. The A5, for example, follows an ancient Roman road. All the towns whose name ends in '-cester' (Cirencester, Bicester, Gloucester etc) come from the Latin 'castra' meaning 'army camp'.
We use Latin terms without thinking in our daily language: 'etcetera' meaning 'and the rest', or 'e.g.', standing for 'exempli gratia', which means 'for example'. The Norman invasion brought a huge further influx of Latin and Latin-derived words into English. As a result, about half of the words in our language come from Latin. To illustrate the principle, and to demonstrate how dependent we are upon Latin, every word in this sentence which has a Latin origin is coloured purple.
The Greeks invented western democracy (a Greek word meaning 'rule by the people'); the Romans developed other political and legal concepts that are still in use. Many of the most important philosophical ideas originated with Plato, Aristotle and Roman writers. Virtually every major literary genre history, philosophy, oratory, love poetry, epic, satire, comedy, tragedy was developed in the classical world.
European thinkers and writers built on this heritage, some consciously, others not. Shakespeare and Molière, Dante and Dryden would not have written what they wrote had the Greek and Roman authors not prepared the way. Winston Churchill used to brilliant effect public speaking techniques long ago perfected by Demosthenes and Cicero. More recently, Philip Pullman's trilogy, His Dark Materials, intentionally incorporates themes from Homer and Aeschylus.
Speak and write better English (and French, and German, and Italian, and Spanish...)
Latin is an unforgiving language. Where English tolerates sloppiness in word order and grammar, Latin demands discipline. It is possible to make yourself understood in English without ever knowing the difference between a noun and a verb, let alone between a subject and an object in a sentence. But making yourself understood is not the same as writing elegantly or persuasively. Avoiding ambiguity involves knowing the structure of your language. This is not mere pedantry. If communication is a key skill in an increasingly media-driven society, then those who know how to express themselves well will have a head start.
The same goes for learning foreign languages. If you want simply to speak and read French, then learning French (ideally in France) is the way to do it. But if you want the capacity to learn any major European language quickly, a grounding in Latin will pay handsome dividends. Not only do all of them use the same grammatical and syntactical structure, but many word stems are shared.
Take, as an example, the Latin word 'insula', which means 'island' (and also, by extension, 'block of flats' yes, the Romans had those). Derived from it are German insel, Italian isola, Spanish isla, French île (the circumflex marks a lost 's'), Portuguese ilha, and of course English isle and island. Furthermore, it gives us the English verb to insulate, meaning to separate like an island; and, through the Italian derivation, to isolate.
If Latin imposes discipline, Greek teaches precision. Ancient Greek had an enormous vocabulary, and a tremendous capacity for inventing new words. Choosing the right word to match an English concept is an intriguing exercise: there will almost certainly be several possibilities, each with its own nuance. What other language do you know that has a word for 'to grow into pig-hood'?
And there is more: a normal Greek verb has 812 different forms, each of which means something very slightly, almost imperceptibly different from the other 811. Don't worry: they fit into very logical patterns, and you don't need to know all of them, or even most, for GCSE. But choosing the right one, or figuring out why the author you are reading chose one over another, will awaken you to the possibilities of your own language. Indeed, there is arguably only one major European language with a vocabulary as subtle and as flexible as that of Ancient Greek, and that is English.
Impress your friends, and more
Now that most schools don't teach Latin or Greek, becoming a classicist initiates you into a select group. You will be able to point out where the major Roman settlements were in England, and why they were chosen. You will know when to use 'who' and when to use 'whom' in a sentence. You will have no difficulty spelling accommodate (knowing, as you will, that it comes from Latin 'ad'+'commodus'). You will know that a.m. and p.m. stand for ante and post meridiem (before and after noon), and that therefore 12 noon by definition can't be a.m. or p.m.
You will know the difference between boy's and boys' (genitive singular and genitive plural forms in English). If a doctor says you may suffer from orthostatic hypotension, you will guess correctly that he means you may suffer a drop in blood pressure when you stand up. You will know without looking it up what a chronometer is and what a palaeontologist does. When the Leader of the Opposition says in a newspaper interview (as he did not long ago) that he wouldn't 'resile' from a position he had taken, you, knowing that Latin 'resilio' means to jump back, will understand that he means he's not going to retreat.
At your fingertips will be tags such as 'omnia vincit amor' (love conquers all) and 'fortis fortuna adiuvat' (fortune favours the brave). You will have anecdotes about Socrates, who lectured to his pupils even as he drank poison in obedience to the court's sentence; and Manlius Torquatus, who condemned his own son to death for disobedience in battle (he had fought a duel with an enemy leader, and won but against orders) ...
What are you waiting for?
Why learn Greek and Latin?