Travelling by train is quick, cheap and easy.
Register with http://www.ns.nl/en and download their app, then you can store all of your advance tickets and check train details on your phone.
Yes, we know it's in Belgium, but it's right on the border and you might as well stop in on the way past. Between the station and the river Schledt is a large pedestrianised area with plenty to see, do and eat.
Onze-Lieve-Vrouwekathedraal (Cathedral of Our Lady), north of the large square Groenplaats
There's also impressive architecture in the Grote Markt.
Rubenshuis has many of the artist's paintings, and exhibitions about his links with the city, plus a large garden.
As you're still in Belgium, get some frites and waffles, maybe from a vendor down by the river.
Only one building in Rotterdam survived World War 2, after the Germans razed it to the ground. This means that it has some of the most innovative and stunning architecture in the Netherlands. Not all is good, and it is a major port before it is a tourist city, but by land and sea there is a lot to discover.
Hilton Rotterdam: only 8 minutes walk from the station, large rooms with enormous windowsills, and very little noise given that it's in the centre. The executive lounge is unusually on the ground floor, and serves a good hot breakfast selection.
Walk along the Westersingel canal down to the waterfront. You can see the Swan Bridge (Erasmusbrug) and many modern skyscrapers.
Just under the north end of the bridge, buy a ticket for the Spido boat tour (https://www.spido.nl/en/tochten-cruises/rotterdam-havenrondvaart), 75 minutes with a commentary in English showing you the industry and design in the port. Up to 10 tours a day, €12.50.
Watch out for the Euromast, a tower with revolving restaurant that you can visit if so inclined. You also get to see at least one windmill.
The maritime museum is on the way back to the centre, and has many outdoor boats you can visit for free, as well as exhibitions inside.
You will also want to see the White House, the only building from 1898 in the city, just along the canal.
Markthal is the most amazing market hall we have seen. While keeping the traditional food stalls, and mixture of cafes and restaurants, it's set in a giant glass and metal dome that is stunning whatever the time of day.
The Cube Houses are visible from the Markthal. Designed by Piet Blom, they are literally cubes but upended onto their points and making a bridge over the main road. One is open as a museum and you can explore every room and see the original 1980s furnishings. http://www.kubuswoning.nl/introkubuseng.html
For culture, the Museum Park is a short walk from the centre, and has the Kunsthal park and garden, the Boijmans art museum, and a natural history museum. In the area are several 1960s Brutalist houses, white concrete with flat roofs.
Other attractions: Tax Museum, De Hef transporter bridge, the Fotomuseum and the Cathedral.
South of the station is Lijnbaan, a shopping district reminiscent of Hatfield, or other "new towns" of that era.
In the Markthal, we recommend Elliniko (http://www.elliniko.nl) for their Greek food, including a giant platter of meat.
Vapiano is a great Italian restaurant where the pasta is cooked in front of you, we went to the one in the Plaza near the station.
If you haven't had Stoopwafels yet, buy some at an Albert Heijn. Delicious as snacking biscuits, or lovely warmed over a cup of hot chocolate.
A short ride from Rotterdam, the train deposits you on a windswept plaza with views of a tram and some scummy canals. Don't be put off! This beautiful historic town is only one block away.
Oude Kerk and Nieuwe Kerk: only in Delft could the "new church" date from 1351. The old church goes back to 1240, or even earlier. Both are worth visiting, the old down a quiet side street, and the new facing onto the Markt, opposite the Town Hall.
Vermeer Centre: While it doesn't feature any of his 37 original works, this building that Vermeer regularly visited has an exhibition of his life and influence. Around the town are rotating cubes that give more information on the locations where Delft lived, worked and painted.
De Diamanten Ring, "sinds 1796", is the oldest bakery in Delft. Now is your chance to try Dutch delicacies such as butter cakes, Scheve Jantjes, meringues, and Willem van Oranje-brood.
There are many shops on the Markt selling cheeses, clogs, cheeses in the shape of clogs, windmills, and Delft pottery made in China. Don't get fooled!
aka The Hague, home of the International Criminal Court, and MC Escher. Again you can see this easily in a day, unless you want to visit Scheveningen, Den Haag's seaside resort, which is lovely in summer (and bracing in winter).
One of the best tourism websites we have ever seen: https://denhaag.com/en
Escher In Het Paleis: the optical work of the woodcut, lithograph and mezzotint artist whose works have baffled and inspired many.
Panorama Mesdag: another of the surviving great European panoramas of the 19th century, this one unusually shows a coastal scene, Scheveningen in 1881.
Mauritshuis: Girl With A Pearl Earring. Say no more.
Gemeentemuseum: Mondriaan's work, plus many others, in a modernist red-brick gallery.
Once you're there, think about getting an Amsterdam City Card, with free public transport, a free canal cruise (4 companies on offer), free entry to most museums (not Rijksmuseum or Anne Frank Huis), and discounts on other attractions and food. €57 for 24 hours up to €77 for 72 hours.
't hotel (short for Het Hotel, i.e. the hotel)
This tiny 8 room canalside haven is very easy to get to, and lives above an antique shop. The breakfast bar is in the basement, but light and nicely decorated. Some rooms look out onto the canal.
Pancake Bakery, Prinsengracht 191 (http://www.pancake.nl/en/)
Semhar, Marnix Straat 259: an Eritrean restaurant with great dishes including enjera (plates made of pancakes covered in meat) and vegetarian and seafood ones (http://semhar.nl/)
Amsterdam Historical Museum, Kalverstraat 92
Anne Frank Huis, Prinsengracht 267
Van Gogh Museum, Paulus Potterstraat 7
Rijksmuseum, Stadhouderskade 42
Jewish Historical Museum, Jonas Daniel, Meijerplein 2-4
The home of Miffy (or Nijntje in the original Dutch). You can't get away from her. The little rabbit created by Dick Bruna has her own traffic lights (at a rainbow pedestrian crossing, LGBT fans - on Lange Viestraat, outside de Bijenkorf), her own museum (opposite the Centraal Museum), and numerous bookshops have her stories in 8 Dutch dialects (and Latin).
Centraal Museum: Dick Bruna was a book illustrator and graphic artist as well as designing Miffy. The museum has a recreation of his studio, and film of him talking about his work. It also has a very nice restaurant.
Dom Under: an innovative combination of archaeology, history and fun. After an orientation talk about the history of Utrecht, from Roman times to the hurricane that divided the cathedral, you are given a torch and taken to explore the darkened ruins beneath. As your light shines on an artefact, the commentary begins in your headphones. Cue much waving torches madly looking for sensors!
Dogma: gourmet hot dogs served in a burrito-style cafe (http://dogmahotdogs.com/)
The home of Philips electronics company, this city has a Philips Museum with designs for bulbs, radios, shavers, personal stereos and medical equipment dating back to 1891. Definitely worth breaking your journey for (which is perfectly OK in the Netherlands, by the way).
Though it's a long way from most other Dutch cities, Maastricht has a regional feel and is set apart in ways other than geography. Also, because it's so close to the Belgian border, taking the train back to the UK can be made cheaper by getting a cheap ticket to Liège, and then using your "Any Belgian station" Eurostar ticket from there back home, stopping off in Brussels for a bit on the way.
Boekhandel Dominicanen bookshop: combining a sight, a shop and an eat, this is the number one destination in Maastricht. The interior has been transformed, the cafe turns a sanctuary into a cruciform table, and the books (while mainly in Dutch) are food for the soul. https://www.libris.nl/dominicanen
De Bisschops Molen: a bakery with its own waterwheel, using only spelt flour (http://www.bisschopsmolen.nl/). They do delicious rice balls (with fruit), honey buns, platsen, knapkoeken, and more.
Out of town:
Trenches, shelters, grottos and tunnels. We're saving them up for our next visit, but you can check them out here: http://www.maastrichtunderground.nl/eng/caves