Eight days in Wales and Ireland: castles, a mountain railway and the boat across

A Roman fort, a Norman motte and, wrapped around them, the apartments the third Marquess of Bute had William Burges redesign from 1868: a gilded Arab room, a nursery tiled with fairy tales, a roof garden. The house tour is separate from the general ticket and is the reason to come.

The castle's old grounds, running north from the back wall along the river Taff as an arboretum and open parkland. There is a stone circle put up for an eisteddfod, and the walk continues out of the city if you keep going.

A cast iron and glass market hall of 1891 with a gallery running round the upper floor, selling fish, records, Welsh cakes cooked on a griddle in front of you and, on one stall, live pets. It closes late afternoon, so this is the last stop rather than the first.

A Victorian pub on a corner off St Mary Street, all dark wood and tiled floor, which fills with rugby crowds on international days and is calm on every other one. Ask what the local cask is.

An open-air museum of more than forty re-erected buildings in the grounds of an Elizabethan manor, including a medieval church repainted as it would have looked, a working smithy and a terrace of six miners' cottages each fitted out for a different generation. Free entry, charged parking, and half a day at least.

The Mermaid Quay side of the bay is all chains; walk two minutes to the Norwegian Church or up towards Bute Street for something better and cheaper.

A bronze-coloured shell of Welsh slate and steel opened in 2004, with a line of poetry cut right through the front in letters two metres high, in Welsh and English, lit from behind at night. Free to walk into, and there is usually something free happening in the foyer.

The Welsh parliament building, opened on St David's Day 2006, with a slate floor, a wave of Welsh oak for a ceiling and a public gallery you can walk into without booking when the Senedd is sitting. Free, and deliberately transparent in the literal sense.

Roughly four hours, changing once, then a short hop or a fifteen minute walk from Llandudno Junction across the estuary into Conwy. Book ahead; walk-up fares on this route are punishing.

Built for Edward I between 1283 and 1287 on a rock above the estuary, with eight towers and no keep because the rock does the work. The great hall is roofless and the wall walk gives you the whole town, the river and the mountains behind in one turn.

A merchant's town house begun in 1576 with its original plasterwork intact: heraldic overmantels, a painted screen and a kitchen fitted out as it would have been. It is one of the best preserved Elizabethan townhouses anywhere in Britain and takes about an hour.

A red terraced house on the quay measuring six feet across and ten feet high, lived in until 1900 when the council condemned it, allegedly with a six foot three fisherman as its last tenant. It takes ninety seconds and it is in the record books, which is the whole joke.

A pub on the quay with tables outside facing the estuary and the castle behind you. Conwy mussels are landed twenty metres away and are on the menu in season.

A cable-hauled street tramway of 1902, one of only three left in the world, hauling wooden cars up a one in four gradient from the middle of town to the summit in two stages. It runs from late March to October and closes for the winter.

A bronze age copper mine on the headland, rediscovered in 1987 and now the largest prehistoric mine open anywhere: several kilometres of tunnels cut with stone hammers and bone tools around 1800 BC, some so narrow that only children can have dug them. Hard hat provided, and it is genuinely tight in places.

Llandudno does the seaside properly: chips on the front, ice cream, and a couple of good kitchens on Mostyn Street a block back from the water.

The longest pier in Wales at almost seven hundred metres, opened in 1878, with the original ironwork and kiosks still on it and a bandstand at the far end. Free to walk, windy in any weather, and the view back at the town is the reason to go to the end.

The only rack and pinion railway in Britain, opened in 1896 and climbing 950 metres in under five miles to a station just below the summit of Snowdon. It takes about an hour each way with half an hour at the top, and the steam-hauled services are worth the surcharge.

The village high street has cafes built for walkers, which means large portions early and no interest in anybody's muddy boots.

A round stone tower on a knoll between two lakes, built by Llywelyn the Great in the 1220s and one of the few castles in Wales built by the Welsh rather than against them. It is a ten minute walk up from the village and free.

The country park on the old Dinorwig quarry terraces above the lake, which was one of the largest slate workings in the world and is now part of a UNESCO listing. The walk up the incline past the abandoned levels takes two hours and the scale of what was cut out of the hillside is the point.

Fifty minutes on the train, then three hours fifteen on the conventional sailing. Foot passengers walk on and off. Sit on the open deck as you leave: Holyhead Mountain and the South Stack lighthouse are the last of Wales you see.

A cast iron footbridge of 1816, the first over the Liffey and for over a century the only one, named for the toll charged to cross it. It is the shape everybody associates with Dublin and it takes a minute to walk.

A low-ceilinged pub off Bridge Street claiming a licence back to 1198, with a courtyard and trad sessions most nights. The claim is arguable and the room is not.

Temple Bar is two minutes away and charges for the atmosphere. South William Street and Fade Street are five minutes further and much better for the money.

A walled Georgian campus founded in 1592 and still a working university, with a cobbled front square you can walk into free. The Book of Kells, an illuminated gospel of about the year 800, is in the Old Library, which is midway through a long restoration, so check what is open before buying.

The seat of British administration in Ireland until it was handed over in 1922. The tour takes in the medieval undercroft and the Chapel Royal; the State Apartments you walk yourself, and the free Chester Beatty Library in the garden behind is quietly one of the best small collections in Europe.

The road running west from the castle towards the brewery has been the way into Dublin for a thousand years and now feeds students and market traders. Quick, cheap and not aimed at visitors.

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 tour, and the included pint in the glass-walled bar at the top is the best view in Dublin.

Free, and holding almost everything important dug out of Irish ground: the Ardagh Chalice, the Tara Brooch, hoards of bronze age gold, and four iron age bog bodies displayed with more care and less sensationalism than you expect.

About half an hour from O'Connell Street or the quays, several an hour, and a fraction of a taxi. Buy on board or in the app.