Londoners couldn’t believe their luck when Tate Modern first opened back in the year 2000. Here was a space so big and multi-functional you felt you could pretty much live in it: wafting between its galleries, cafés and temporary exhibitions, soaking up the views towards St Paul’s, meeting friends — and generally blurring the lines between gallery-going and the rest of life.

London is home to three of the world’s top schools of music — the Royal Academy, the Royal College and the Guildhall — and public performances are a key part of their study programmes. So why not offer yourself as a guinea pig and enjoy a no-cost concert by the future stars of the classical scene? Even though they’re free, events are ticketed — so you’ll need to book in advance.

Quick, go and see the Elgin Marbles before public opinion turns decisively in favour of their rightful return to Athens. And, while you’re at it, clear your diary for the rest of the day. The BM offers a matchless introduction to the birth of art and high culture around the world.

For kids, nothing washes away the boredom of hot city afternoons better than jets of water squirting out of the pavement. London has several of these joyous, public-spirited fountains, and not surprisingly they’ve become a staple of summer life in the city.

Prime Minister’s Questions is the star attraction and takes place at midday on Wednesdays when the Houses of Parliament are sitting. Entry is free and ticketed: but restricted to UK residents (you need to apply to your local MP for them). But for the rest of the week anyone can queue for entry via the Cromwell Green entrance.

Lately, the indoor Sky Garden has become a must-do on many London visit wish lists — so much so that you have to book entry to the office block on which it rests weeks in advance. But what few visitors realise is that there’s another rooftop garden only 200 yards away, and on a sunny day it’s even lovelier because it’s open to the sky.

Don’t try to see it all at once. The National Gallery is so big — and its collection of pre-20th century painting so broad and magnificent — that your art-loving brain will have melted by the time you’re half-way through. So why not nibble at it instead? It’s free after all.

Admittedly, you’ll need a budget to buy the odd irresistible item: whether it’s a sourdough loaf from Bread Ahead, or one of one Brindisa’s chorizo rolls. But otherwise Borough Market — the spiritual home of London’s foodie scene — is free.

Mithraeums were temples to the Persian god Mithras: and in Roman times they were popular — if mysterious — places of worship among the Empire’s ruling elite. This example was unearthed in London in 1954 and, after a spell in the open air, was reinterred 23 feet beneath Bloomberg’s European HQ.

Don’t fancy the eye-watering price of entry to St Paul’s? Then walk for free into St Stephen Walbrook, between Mansion House and the London Mithraeum. Some fragments of the 15th-century original remain, but the sense of wonder here is almost entirely down to the building Sir Christopher Wren designed after the Great Fire of London.

Have you been to Victoria Park lately? For several years now it’s been a window onto east London’s hipster rebirth, and the scene here is more local and — many would argue — more vital than the park life further west.

The Central Criminal Court of England and Wales — aka the Old Bailey — is where many of the most hair-raising cases in the country are heard. And if you’ve got a stomach for the details, and an interest in the legal process, then a few hours in the public galleries are a must.

On Saturday, this long, narrow road in Notting Hill — packed with stalls and indoor arcades — hosts the UK’s biggest antiques market. Years ago, it morphed into a big, chaotic promenade for the city’s chattering classes — full of street food, curiosities and chance encounters with friends.

This long, narrow stretch of woodland — which dates back at least 3,000 years — serves up London’s largest continuous stretch of greenery, as well as its loveliest natural landscapes.

Never mind what’s on. The experience of seeking out the White Cube gallery is as much fun as the art itself. It takes you from London Bridge station, beneath the soaring spike of the Shard, and into the low-key Bermondsey Street.

Contemporary art doesn’t just hang in galleries. So head to the Leake Street tunnel, beneath the railway lines of Waterloo station near the South Bank, for 300m of graffiti, tags and street art.

It’s hard not to get excited as you exit South Kensington Tube station and approach London’s biggest museum hub. Three vast palaces of fascination await — the Victoria & Albert, the Science and the Natural History Museums — and it’s best to treat them as a giant box of pick and mix treats, rather than three separate institutions.

Fuelled by punchy lattes and cream cheese bagels from Café Columbia, this weekly gathering of florists and potted plant specialists offers something few other London freebies can match: a sense of home.

The standard London Good Delivery gold bar weighs 11-13kg and you can try lifting one yourself courtesy of the Bank of England. You’ll find it in the Bank’s free museum, squirreled away near its eastern entrance.

Don’t fancy the crowds at Buckingham Palace? Or the fact that the key ceremony during the changing of the guard takes place behind iron railings? Then go to Horse Guards Parade instead. At 11am each weekday, the King’s Life Guards, on duty at Horse Guards on Whitehall, are relieved by a new detail of cavalrymen, who have ridden down from Hyde Park Barracks.