Pictured: The Sky Garden, 34 storeys up, has 360-degree views across London. (See No 27 for full info)
1. Visit a Farmer’s Market
Browse and buy fresh, local produce, or try new local foodstuff from the stalls. Producers, from fisherman to farmers, and growers to bakers usually have a regular outlet near you in most parts of the country. Farma.org.uk, which is searchable by postcode, lists farmers markets across the country.
2. Free guided walks in the Lake District
Throughout the month
The National Park offers a range of guided walks for all abilities: on 17 May, for instance, you can discover Wordsworth’s Hawkshead in a short 4km walk, being run in conjunction with the Wordsworth Museum. It’s free, but donations are welcome.
3. Go through the longest cycle tunnel in Britain
Near Bath, Avon
Opened in 2013, the Two Tunnels cycle path, built on disused rail-lines, offers a range of route-options for all abilities, exploring the countryside around Bath.
It includes the Combe Down tunnel, the longest cycle tunnel in Britain – it’s over a mile long, but features an interactive light and sound installation to help the yards fly by.
4. 50 things to do before you're 11 and three-quarters
Countryside across UK
The National Trust protects and runs more than 350 historic houses, gardens and ancient monuments, forests, woods, fens, beaches, farmland, downs, moorland, islands, archaeological remains, castles, nature reserves, villages – many of which are free to visit.
Its project '50 things to do before you're 11-and-three-quarters' is just that: a project for kids to sign up to, offering ideas for free outdoors activities. You might start by flying a kite at White Horse Hill, for instance...
5. Scottish National Gallery
The Mound, Edinburgh, EH2.
The Scottish National Gallery is in the heart of Edinburgh and houses one of the best collections of fine art in the world. Admission is free. A charge may be made for special exhibitions.
6. Live folk gig on a train
You read that right: the Folk Train runs from Sheffield at 19.14 on the fourth Tuesday of every month, with a live band playing a free show all the way to Edale in the Peak District (19.47). There, everyone gets off and goes to the station pub, where the session continues until they catch the 21.28 train back to Sheffield. There's also a service from Manchester. folktrain.org.uk
7. Watch your own skeleton ride a bicycle
Museum of Science and Industry, Liverpool Road, Manchester M3.
Stare into the mirror of infinity, create a tornado, and - yes – watch your own skeleton ride a bicycle at Manchester's brilliant MSI. There's 25 hands-on exhibits to play with and see science in action. Find out how a mill engine works, and discover its role in the Industrial Revolution. there's also a programme of regular special events and talks, many also for free.
8. Take a free tour of Bath
Follow in the footsteps of Jane Austen by taking a walking tour around Bath. Visitors can download a free MP3 recording which allows you to trace the footsteps of one of Bath's most famous residents.
The tour takes you around the highlights of the city and includes extracts from Jane Austen's novels and letters, which describe Bath as it would have been in its Georgian heyday. Download the MP3 here
9. Late night at the Science Museum
Exhibition Road, London SW7.
'Lates' at the Science Museum is a free night for adults that takes place on the last Wednesday of the month, themed around a different scientific subject each time. Alongside events and talks, 'Lates' have live music and other entertainments, sometimes including a silent disco... It's a great chance to see one of London's great institutions in a completely new light.
10. Late at Tate
Tate Britain, Millbank, London SW1P.
The Tate Britain comes alive in a new style on the first Friday of alternate months – August, October, December 2016 – as the galleries play host to music, performance, talks and events, between 6pm and 10pm. There's usually a great atmosphere, it's all free and it feels like a real London secret.
11. Explore the history of Lincoln
The Museum of Lincolnshire Life’s rich and varied social history collection reflects and celebrates the culture of Lincolnshire and its people from 1750 to the present day. Exhibits illustrate commercial, domestic, agricultural, industrial and community life.
12. Immerse yourself in football
National Football Museum, Manchester M4 3BG, 0161 605 8200
Huge, interactive celebration of British and international football, with a changing roster of temporary exhibitions, plus regular football training and other activities for kids.
13. Marconi radio museum
Sundays 1.30-4.30; Tues and Fridays. 7-9pm. Mullion, Cornwall
Small but acclaimed museum high on the Cornish cliffs that celebrates the first-ever Transatlantic radio signal - sent to Marconi in Canada from this spot in 1901.
Staffed by enthusiastic local volunteers, the Centre will give you an insight into one of mankind's greatest inventions - and you'll get the chance to send some Morse code yourself. It's not open every day, so be sure to check the website before you set out.
14. Immerse yourself in architecture
Narrow Quay, Bristol BS1
Bristol’s Architecture Centre is an independent, not-for-profit organisation that champions better buildings and places for everyone, with a series of exhibitions, talks, visits and other events. the gallery is open to the public five days a week (Wednesday to Sunday) and entry is free.
15. See Free Music in Birmingham
Fridays, 5pm, Birmingham Town Hall
Get that Friday feeling and check out the free Jazzlines session at Birmingham's Town Hall Symphony Hall every Friday at 5pm.
For the full programme of shows, see: thsh.co.uk/jazzlines
16. Enjoy Sheffield's huge indoor garden
Sheffield Winter Garden, Millennium Square, Sheffield. 8am-6pm every day
If you fancy stretching your legs but it’s too cold for a long walk outside, visit Sheffield's impressive multi award-winning Winter Garden - one of the largest temperate glasshouses to be built in the UK in the last 100 years. More than 2,500 plants from around the world, the Winter Garden is an oasis in the heart of England's fourth largest city.
17. See the other side
Speakers' Corner, North-east edge of Hyde Park, London, close to Marble Arch
Marx, Lenin and Orwell all spoke here at Speakers' Corner after an 1872 act of parliament had ratified the use of this part of Hyde Park for public speaking. Even now, in the digital age, it's not unusual to find crowds gathering at Speakers' Corner to listen to enthusiasts expounding their views.
Anyone can turn up unannounced to speak on any subject, as long as the police consider their speeches lawful.
18. Yorkshire Sculpture Park
West Bretton, Wakefield, WF4.
Huge open-air showcase for the work of many of the world's great sculptors: a great place for an afternoon walk, through the woods and by the lake, with the added value of the landscape being scattered with the work of Henry Moore, Barbara Hepworth and an ever-changing roster of British and international artists. And there's a cafe, too.
19. Margam Country Park
Neath, Port Talbot
A magnificent castle, 18th-century orangery, ornamental gardens and deer park are set in 1,000 acres of lovely countryside, with attractions including a narrow gauge railway, adventure playgrounds, nursery rhyme-themed children’s area, and a rare breeds farm trail.
Entry to the park itself and most of its attractions are free. The park hosts various events from Living History Weekends to Clown Festivals and Charity Walks.
20. Relax in an old-fashioned living room
Geffrye Museum, 136 Kingsland Road, London, E2 8EA
Focusing on the history of the British living room, the Geffrye Museum tells a fascinating story of fashion, taste and social change. There’s also an airy restaurant overlooking the lovely gardens, which include a series of plots in different historical styles.
21. Free early-morning workouts by the River Thames
The Scoop, Queen's Walk, London SE1
Every Wednesday morning at 6.30am, right next to Tower Bridge, you can take part in a mass workout for free, thanks to Project Awesome. It's designed to motivate and encourage people of all ages, shapes, sizes and fitness levels to get outside, sweating and smiling. At silly o'clock. Time Out calls it 'London's friendliest workout'. Let us know how you get on.
22. Get in the face of St Paul's Cathedral
One New Change, London EC4M
Take the lift to the top of the shopping centre and discover a roof terrace with a spectacular view – with St Paul's Cathedral almost within touching distance. The Roof Terrace is open to the public seven days a week from 6am to midnight. It's free - and there's plenty of space to enjoy your own a picnic up there – but there's also a restaurant and bar if you want to make an afternoon or evening of it.
23. A long tunnel in the middle of the countryside
Standedge Tunnel & Visitor centre
In the heart of the Pennine countryside, Standedge is the longest canal tunnel in Britain and makes a great family day out, with free entry to the visitor centre and regular falconry displays and boat trips through the tunnel.
24. Modern art in the middle of Wales
Wales Museum of Modern Art, Machynlleth, SY20 8AJ
From the front, it looks about as big as a corner shop – but MoMA opens up into a hidden world of many galleries, with a lively roster of permanent and temporary exhibitions. Always worth a stop.
25. See where your water comes from (if you live in, or near, Birmingham)
Elan Valley Visitor Centre, Rhayader, Powys LD6 5HP
The best of all worlds: Elan Valley is a haven for wildlife in the Cambrian mountains in mid-Wales and full of magnificent views - but it also has a fascinating part in the history of modern, urban, Britain: since 1896, water has been moved from here along a 73-mile pipeline to the city of Birmingham.
See the remarkable story of the dams and railways and aqueducts that brought that project to fruition, have a walk in the country – and a cup of tea. There's a children's playground too.
26. London's Roman Amphitheatre
Under Guildhall, London EC2V
Beneath the courtyard at the front of the Guild Hall in the City of London lie the ruins of London's Roman Amphitheatre, where thousands once gathered to watch animal fights, public executions and gladiatorial combats.
The little-known site is bolstered by museum-style interpretations and is open daily, for free. Afterwards, you can take in the recently refurbished Guildhall Gallery, another hidden London gem, also free.
27. Enjoy a secret garden, 34 storeys up
The Sky Garden, 20 Fenchurch Street, London EC3M
At the top of the 34-storey skyscraper known as the 'Walkie Talkie', there's a hidden gem: a garden with 360-degree views across the city. There are bars and restaurants up here too for cocktails and more as the sun goes down. The Sky Garden is free to enter, but you need to book in advance.
28. Warriors through the ages
Royal Armouries Museum, Leeds, LS10 1LT
See arms and armour of warriors through the ages and around the world, and watch live combat demonstrations and dramatic performances with a lively programme of daily events. There are five galleries to explore – War, Tournament, Oriental, Self Defence, Hunting – and of course the Hall of Steel.
29. See the world under one roof
Great North Museum, Newcastle, NE2 4PT
Experience a planetarium, see ancient mummies, dinosaur skeletons, a rare Inuit canoe, and a scale model of Hadrian's Wall. Since its £26 million refurbishment in 2009, Newcastle's huge interactive museum has welcomed over 1 million visitors.
30. Catch a public lecture
London School of Economics and Political Science
The LSE runs a packed program of talks on a variety of subjects – mostly free to the general public. You can also sign up to live podcasts if you can’t make a specific lecture you like the sound of, and there's a free annual literary festival of talks on a wide range of subjects too, every February.