Sixteen days in the British Isles: six cities and one crossing of the Irish Sea

Nelson up his column with Landseer's four bronze lions round the base, and a fourth plinth in the north west corner carrying whatever contemporary sculpture is on rotation. The National Gallery fills the north side and is free, which makes it the default wet-afternoon plan.

The old wholesale fruit and vegetable market, closed in 1974 and now an iron and glass arcade with buskers auditioned by the estate working the piazza. The theatre district starts at the edge of it, so this is where to see what is on this week.

Ten minutes north and you are in the densest restaurant streets in Britain. Walk in early, because the queues start at seven and most of the good places do not take bookings for two.

Started by William the Conqueror in 1066 and never left alone since: a fortress, a mint, a menagerie, a prison, and the vault for the Crown Jewels, which you view from a moving walkway so the queue never stops. The White Tower and the ravens are the parts people remember.

Victorian, hydraulic and still raised several hundred times a year for tall ships, with a high walkway between the towers that has a glass floor in it. The road deck costs nothing and is where the photograph is taken from anyway.

The commuter clipper from Tower Pier takes about twenty five minutes and takes contactless payment at the gate. Sit at the back, outside, and watch Canary Wharf come up on the left.

Founded by Charles II in 1675 to fix the problem of longitude at sea, and the line on the ground here is the zero from which every other place on earth is measured east or west. The Harrison sea clocks are inside and are the best objects in the building. The hill it stands on is the view.

Free and inexhaustible, with the Rosetta Stone, the Parthenon sculptures and the Assyrian reliefs in the first twenty minutes and the arguments about how they got here printed on the labels. The Montague Place entrance at the back has no queue.

The squares behind the museum feed a university, so the coffee is good and the sandwiches are not museum-priced. Russell Square has benches and a fountain.

Coronations since 1066, and beneath the floor a substantial proportion of the people who ran the place afterwards: poets in the south transept, Newton and Darwin in the nave, and the Unknown Warrior just inside the west door. Ticketed by time slot, or free if you come to a service.

Barry and Pugin's Gothic revival parliament, built after the 1834 fire around the surviving Westminster Hall, which dates to 1097 and has a hammerbeam roof worth the tour on its own. Tours run Saturdays and in recess; the rest of the time, look at it from the far end of the bridge.

About two and a quarter hours on the West Coast Main Line, roughly hourly. Lime Street station puts you ten minutes' walk from the waterfront and five from most of the hotels.

A ring of brick and cast iron warehouses of 1846, built without a scrap of structural timber so the cargo could not burn, and now the largest collection of Grade I listed buildings in the country. The water in the middle is still tidal dock.

Free, and much sharper than a civic museum has any right to be: the port, the Irish and Welsh migration, the football, the music, and an honest section on where the money came from. The upstairs windows frame the waterfront better than any viewpoint in the city.

A fifty minute river cruise across to Birkenhead and back with commentary, and yes, they play the song. It is the only way to see the three buildings on the Pier Head the way they were designed to be seen, which is from the water arriving.

A rebuilt version of the cellar on Mathew Street where the Beatles played almost three hundred times before anybody outside Liverpool had heard of them. The original was filled in during the 1970s. Live music from lunchtime, and it is not remotely a museum piece.

A Greco-Roman hall of 1854 with a tunnel-vaulted concert room and a Minton tile floor of thirty thousand tiles that is uncovered only a few times a year. Widely called the finest neoclassical building in Europe, which for a provincial concert hall is quite a claim.

Free, and one of the best collections outside London: Rembrandt, Poussin, a strong Pre-Raphaelite room and Hockney's early work. Opened in 1877 and paid for by a brewer, which was as usual how these things happened.

Ten minutes south, and the best eating street in the city: Middle Eastern, Caribbean, Italian and a bakery queue that moves faster than it looks.

The largest cathedral in Britain and one of the largest anywhere, in red sandstone, begun in 1904 and not finished until 1978. The tower is 101 metres and you go up it in two lifts and 108 steps for the best view on Merseyside.

Three and a half hours, usually direct, occasionally with a change at Preston or Carlisle. The stretch over the Southern Uplands after Carlisle is the good part.

A 61 metre Gothic rocket over a seated Walter Scott, still black with two hundred years of coal smoke because cleaning it would take the surface off. The climb is 287 steps and the staircase narrows all the way to the top.

A stubby hill at the east end of Princes Street covered in neoclassical monuments, including a Parthenon that ran out of money after twelve columns in 1829 and has been called Edinburgh's disgrace ever since. Ten minutes up, and the whole Old Town ridge laid out in front of you.

A Victorian bar with an island counter, a stained glass gallery upstairs and six Doulton tile portraits of inventors round the walls, Faraday and Watt among them. Stand at the bar, order, and look up.

Eleven hundred years of occupation on a volcanic plug, holding the Honours of Scotland, the Stone of Destiny and St Margaret's Chapel of about 1130, the oldest surviving building in Edinburgh. The gun goes off at one every day but Sunday and everybody still jumps.

The high kirk of Edinburgh, with a crown steeple that has topped the Old Town since the 1490s and where John Knox preached through the Reformation. The Thistle Chapel of 1911 in the south east corner is ludicrously carved and takes two minutes to find.

The Mile itself sells tartan and fudge. Drop down Cockburn Street or into the Grassmarket and the standard rises immediately for the same money.

The King's official residence in Scotland, with the ruined abbey of 1128 attached to the back and the chambers where Mary Queen of Scots was present when her secretary Rizzio was dragged out and killed in 1566. Shuts for about a week each summer when he is in.

The high point of Holyrood Park, with the city, the Firth of Forth and the hills of Fife below it. The path from the Holyrood car park past the loch is the easy way up; the Salisbury Crags route is steeper and better.

Come off the hill at Holyrood and go back up the bottom of the Mile. The cafes around St Mary's Street serve breakfast until noon and you will want all of it.

Free, and the one museum to do if you only do one: a Victorian iron and glass gallery flooded with light, joined to a modern block covering everything from the Lewis chessmen to Dolly the sheep. The roof terrace at the top is public and mostly empty.

The royal family's floating residence from 1954 to 1997, moored in Leith and left exactly as she was decommissioned, down to the sitting room magazines. The audio guide is unexpectedly candid, and the engine room is spotless enough to eat off.

Fifty minutes, four an hour, no reservation needed and cheap enough to buy at the machine. Queen Street comes out directly onto George Square.

The civic centre of the city, laid out in 1781, with a column in the middle carrying Walter Scott rather than George III and twelve other statues arranged around him. The City Chambers on the east side is the grandest thing in Glasgow.

The only medieval cathedral on the Scottish mainland to come through the Reformation with its roof on, dark and tall and built over the tomb of St Mungo in a lower church that is the best space in the city. Free, and run by Historic Environment Scotland.

A Victorian necropolis on the hill behind the cathedral, fifty thousand burials and about three and a half thousand monuments, with John Knox on a column at the top glaring down at the building he would not have approved of. Free, open, and the view back over the city is the reason to climb it.

The grid of old warehouse streets east of George Square is where Glasgow eats and drinks late. Nothing needs booking on a weeknight.

Free, red sandstone, opened in 1901, and the most visited museum in Scotland: Dali's Christ of St John of the Cross, a Spitfire hanging over the west court, armour, Egyptian rooms and a full-size organ that gets played most days at one o'clock.

Ten minutes north into the university quarter, where Ashton Lane is a cobbled alley of bars and kitchens that looks like a film set and mostly is not.

Zaha Hadid's zigzag shed on the Clyde, free, holding trams, locomotives, a recreated 1900s street and a wall of model ships built in the yards a mile downstream. The tall ship moored outside is included.

Kate Cranston's temperance tea rooms of 1903 on Sauchiehall Street, designed inside and out by Mackintosh down to the cutlery and the waitresses' dresses, and restored to what he drew. Tea and a cake in the Salon de Luxe is the point.

Roughly two and a half hours by bus to Cairnryan, then a two hour sailing. The ferry companies sell it as one through ticket. Get out on deck as you leave: Ailsa Craig and the Mull of Kintyre are both visible on a clear day.

White Portland stone and a copper dome, finished in 1906 with money the city had rather too much of. Free tours run daily; the grounds hold a memorial garden listing every Belfast victim of the Titanic, which was built two miles away.

Gas lit, tiled, mirrored and owned by the National Trust, with ten carved wooden snugs down one side that you can shut yourself into and a bell to call the bar. Grab a snug if one is going and look at the ceiling.

Opened in 2012 on the Harland and Wolff slipways, with nine galleries running from the linen mills that made Belfast rich to the wreck on the seabed. The shipyard ride is the best part; the last gallery, with the survivors' testimony, is where the room goes quiet.

The Titanic Hotel next door is the old Harland and Wolff head office, and lunch is served under the barrel-vaulted roof where the ships were actually drawn.

The last White Star Line ship afloat anywhere: a tender built in 1911 to ferry first class passengers out to Titanic at Cherbourg, where the harbour was too shallow for the ship herself. Restored in the dry dock beside the museum and on the same ticket.

Two yellow gantry cranes over the old yard, Goliath from 1969 and the taller Samson from 1974, visible from most of Belfast and now listed as historic monuments. They have not built a ship under them for years, which is the whole story in one image.

About two hours with a driver from one community or the other, stopping at the murals on both roads and at the gates in the peace wall that are still shut at night. Book the day before.

A Victorian radial gaol opened in 1846 and in use until 1996, with a tunnel under the road to the courthouse so prisoners never saw daylight between cell and dock. The condemned cell and the execution chamber are still there and the guides do not soften it.

The university end of the city, which means cheap, fast and open all afternoon, and a short walk from the museum.

Free, and far better than it needs to be: Irish gold, an Egyptian mummy called Takabuti unwrapped in Belfast in 1835, and a floor on the Troubles that presents the arguments without picking one. Two hours does it.

The park the museum sits in, laid out in 1828, with a curved cast iron and glass Palm House begun around 1839 that is among the earliest of its kind. Free, and the first patch of grass in the city to fill up when the sun comes out.

Just over two hours from Belfast Grand Central to Dublin Connolly, about eight a day. Sit on the left for the sea between Newry and Drogheda and the viaduct over the Boyne.

A walled Georgian campus in the centre of the city, founded in 1592 and still teaching. The Book of Kells, an illuminated gospel of about 800, is on show in the Old Library, which is in the middle of a long restoration, so check what is open before buying a ticket.

Nine hectares of Victorian park with a lake, a bandstand and the Fusiliers' Arch at the north corner, five minutes from Trinity's back gate. Locals eat lunch on the grass; you will want to sit down after the Book of Kells.

A narrow Victorian bar with mahogany, mirrors and a snug at the front. No music and no screens, and a pint takes a couple of minutes to settle, so do not panic when it arrives half full.

Temple Bar is where the noise leads and it charges for the privilege. South William Street and Fade Street are five minutes away and much better value.

In use from 1796 to 1924, and where fourteen of the leaders of the 1916 Rising were shot in the stonebreakers' yard in May of that year. The Victorian east wing with its iron galleries is the image everyone knows. Online booking only and it sells out days ahead.

Seven floors inside a converted fermentation plant at St James's Gate, where Arthur Guinness signed a nine thousand year lease in 1759. It is a brand experience rather than a working brewery, and the included pint in the glass-walled bar at the top is the best view in Dublin.

The road between the brewery and the centre has been the way into Dublin from the west for a thousand years and now feeds students and market traders. Cheap and quick.

The seat of British administration in Ireland until it was handed over in 1922, with a medieval undercroft under the courtyard and state apartments upstairs. The Chester Beatty Library in the garden behind is free and quietly one of the best small collections in Europe.

A Viking wooden church stood here around 1030 and the stone one went up in the 1180s, which makes it the oldest structure standing in Dublin. The crypt below is the largest in Ireland and holds, among other things, a mummified cat and rat found inside an organ pipe.

Half an hour from the quays or O'Connell Street, several an hour, and considerably cheaper than a taxi. Buy on board or in the app.