Ten days in Germany: Four cities that barely agree they are the same country

Book the Reichstag dome before you leave home. It is free, it stays open until midnight, and the slots go weeks ahead in summer; you register online with the name and passport of everybody coming.

Start at the gate in the late afternoon. It is the last of the city's eighteen gates, finished in 1791 and modelled on the entrance to the Athenian Acropolis, and it stood inside the death strip for twenty eight years with nobody able to walk through it.

The Holocaust memorial is a block south: 2,711 concrete slabs on rolling ground, unexplained at street level, with a free information centre underneath that is not an afterthought. Then walk up to the river and go up into the dome after dark.

The Wall went up overnight on 13 August 1961 and opened on 9 November 1989. The one place that shows what it actually was is Bernauer Strasse, where a full block of the border strip has been kept: outer wall, raked sand, lamps, a watchtower, and the inner wall on the far side. Houses stood on the line here, and people jumped from upper windows into the west before the flats were bricked up.

It is free, outdoors and takes about ninety minutes with the documentation centre and the viewing platform opposite.

In the afternoon go south to the other half of the story. The Topography of Terror stands on the excavated cellars of the Gestapo and SS headquarters, free and unsparing, with a long stretch of the Wall still standing along the street outside. Checkpoint Charlie is ten minutes further and is now a replica hut in traffic: worth five minutes, and do not pay anybody in costume for a photograph.

Five museums stand on one sandbank in the Spree, built between 1830 and 1930 to put Prussia in the same conversation as London and Paris. The Neues Museum is the one to choose if you choose one: bombed to a shell, left as a ruin for sixty years and rebuilt by David Chipperfield with the damage left visible, so burnt brick meets new concrete without pretending. Nefertiti, carved around 1345 BC, has a domed room to herself.

The cathedral next door was finished in 1905 and the walkway high on its dome is several hundred steps up with the whole island underneath.

In the late afternoon take the S-Bahn east for the other kind of wall. The East Side Gallery is about 1.3 kilometres of the inner wall left standing along the river and painted by artists from all over the world in 1990, and Friedrichshain around it is where to spend the last Berlin evening.

Take a mid-morning train and you are in Leipzig in an hour and a quarter, arriving in the largest railway station in Europe by floor area, built in 1915 with two separate halves because two railway companies refused to share.

Walk into the old town and start at the Thomaskirche. Bach was cantor here from 1723 until he died in 1750, wrote most of his church music in those years, and lies under a bronze slab in the chancel. The boys' choir he directed was founded in 1212 and still sings motets on Friday evenings and Saturday afternoons.

For the evening, find the arcade off the market square. The Maedlerpassage was built in 1914 in glass and marble, and the stairs at one end go down to a wine cellar in business since the sixteenth century which Goethe, a law student in this city, put into Faust.

The Nikolaikirche is plain outside and startling inside, remodelled in the 1780s in pale pink and green with columns that break into carved palm fronds at the top. From 1982 it held peace prayers every Monday at five, which drew a crowd the state could not arrest for praying.

On 9 October 1989 that crowd came out into a square with armed units and water cannon waiting. Something like seventy thousand people filled the ring road holding candles, no order to fire was given, and within a month the government had gone.

Two minutes away is the Round Corner, the Stasi district headquarters, kept exactly as the citizens' committee found it when they occupied the building in December 1989: the offices, the sealed jars of smell samples, the machines for steaming open post. Finish on the market square at the contemporary history museum, which is free and covers the whole East German story.

The three hours to Nuremberg are the longest leg of the trip and the most scenic, running under the Thuringian forest through a chain of tunnels and viaducts built for the line that opened in 2017.

Nuremberg's station sits just outside the medieval wall. Start at St Lorenz, the church facing you as you cross into the old town, which holds Adam Kraft's tabernacle, an 18 metre carved sandstone spire from the 1490s with the sculptor kneeling under one corner of it, and Veit Stoss's Annunciation hanging on ropes above the choir.

Then walk north and up through the lanes to the castle on its sandstone crag. It was never one building but three that spent centuries in rivalry, and the odd highlight is the well, 50 metres of shaft demonstrated by lowering a candle into the dark. Come down for sausages: there is a half-timbered kitchen by the river that has been grilling since 1419.

Tram 8 runs south east to a lake with a colossal brick horseshoe on the far side. The congress hall was designed to seat fifty thousand under a roof that was never built, modelled openly on the Colosseum and abandoned unfinished in 1943, and the documentation centre inside it explains what the rallies were for: the choreography, the film cameras, the trains, and the laws stripping German Jews of citizenship that were announced here in 1935.

Walk on round the water to the Zeppelin field, fifteen minutes away, a long stone grandstand where Speer staged his columns of searchlights. It is deliberately unrestored and the city has argued for years about whether to hold it up.

In the afternoon take the U-Bahn across town to the Palace of Justice. Courtroom 600 is where twenty one surviving leaders of the Nazi state were tried between November 1945 and October 1946, by four powers who had to invent the procedure as they went, and where crimes against humanity were first prosecuted. It is still a working court, so the room closes when a case is sitting; the exhibition above it stays open.

An hour and five minutes on the fast line and you are in Bavaria proper. Drop the bags and go to the middle of the old town, where the neo-Gothic town hall took from 1867 to 1909 to build and looks four centuries older than it is. Its Glockenspiel performs at eleven, at midday and again at five in the warmer half of the year: jousting knights, dancing coopers, about fifteen minutes of it.

For the view, climb the church next door rather than paying for the town hall lift. Alter Peter is the city's oldest parish church and its tower is 306 steps of narrowing staircase with the Alps on the horizon on a clear day.

Walk south through the food market, then east to the Platzl. The Hofbraeuhaus has been on that corner since 1644, holds well over a thousand people downstairs and is full of visitors, which does not stop it being a good evening.

The Wittelsbachs ran Bavaria for seven hundred years and the Residenz is what that looks like from the inside: a palace that swallowed a block of the old town, wrecked in the war and painstakingly put back. Take the museum ticket for the Antiquarium, a barrel-vaulted banqueting hall of the 1570s lined with antique busts, which is worth the entry on its own.

Then go outside for the afternoon, which is what people who live here do. At the southern tip of the English Garden the Eisbach drops over a weir into a standing wave, and surfers in wetsuits queue on the bank to take it one at a time, all year round.

Walk up through the park to the pagoda that was built with it in 1790, and to one of the city's great beer gardens underneath. At the self-service tables you may bring your own food, which is why families arrive with baskets and boards, and the last evening of the trip goes very easily from there.

Finish at the church on the skyline. The Frauenkirche went up in brick between 1469 and 1488 because there was no good stone nearby, and its twin towers were capped with copper onion domes in 1525 when the money for spires ran out. They stand a shade under 99 metres, and a referendum in 2004 made that the height limit for the old town, so nothing central looks down on them.

Inside it is tall, white and plain, rebuilt after the war around the medieval fittings that survived. Then the S-Bahn from Marienplatz reaches the Hauptbahnhof in three minutes and the airport in forty.