Start with the Book of Kells and take the earliest timed slot you can get, because the queue thickens as the morning goes on. Book online in advance; expect to pay somewhere in the region of EUR 25 to 30 for an adult. Know what you are buying before you go. The manuscript is shown downstairs, two open pages at a time, and the Long Room above it is in the middle of a long conservation programme: the great majority of its books, around two hundred thousand of them, have been taken away for cleaning and are not due back for years. In the emptied chamber hangs Gaia, Luke Jerram's seven-metre illuminated globe, which is either a strange consolation or the most memorable thing you will see that hour. A digital exhibition in the Pavilion building finishes the visit.
Five minutes away on Kildare Street is the National Museum of Ireland's archaeology branch, and it is free. This is the collection that makes sense of everything else you will see: the Iron Age bog bodies, the Tara Brooch, the Ardagh Chalice, the Broighter gold. Give it ninety minutes. One planning note that trips people up: on Mondays it typically opens only after lunch, so if Monday is your first day, flip the order and start at Trinity.
Walk on into Merrion Square for the Georgian doors and the Oscar Wilde figure lounging on his rock, then into the National Gallery of Ireland on the square's western side, also free. The Jack B. Yeats rooms and Caravaggio's The Taking of Christ are the two things to find if you have only an hour. It opens seven days a week and stays open late on Thursdays, which makes it a good rainy-evening reserve.
After lunch around South William Street, keep going west. Cut through the courtyards of Dublin Castle to the Chester Beatty, in the castle gardens, which holds early biblical papyri, Islamic and East Asian manuscripts and Chinese jade books. Entry is free with a suggested donation of around EUR 10; Lonely Planet has called it one of the finest museums in Europe, and it is certainly the most underused of the central collections. Then Christ Church Cathedral, then a short walk to St Patrick's, where admission is usually around EUR 11 and Jonathan Swift is buried in the nave he once preached in. Tucked behind it is Marsh's Library, open since 1707 with its original dark oak shelving and the wire cages readers were once locked into with rare books; it opens Tuesday to Saturday and costs around EUR 7.
For the evening, walk back east and drink a street away from Temple Bar. The Long Hall, Kehoe's and Grogan's are all within a few minutes of Grafton Street, and Grogan's toasted sandwich is a legitimate dinner. Walk through Temple Bar to see it, by all means, but pints inside the tourist core run a few euro above the city norm. Camden Street and Wexford Street are where the night goes later.
Day two: Kilmainham first, because the ticket decides your morning
Kilmainham Gaol sells out more reliably than anything else in Dublin. Tickets are typically released about four weeks ahead on the official museum site and Heritage Ireland, and summer slots can be gone within minutes; a small batch of returns is usually put online first thing each morning. No third party is authorised to sell them, whatever a search result tells you. Adult entry is around EUR 8 and the guided tour runs about an hour. The stonebreakers' yard, where the leaders of the 1916 Rising were executed, and the vaulted Victorian East Wing are the images you will carry home. Get there by red line tram or a bus along the quays, roughly half an hour from the centre.
Next door is the Irish Museum of Modern Art, in the seventeenth-century Royal Hospital Kilmainham, free, with formal gardens that almost nobody walks. Give it half an hour or two hours depending on the exhibition.
Walk back east along Thomas Street into the Liberties, and decide about the Guinness Storehouse. It uses dynamic pricing, so expect roughly EUR 26 to 36 depending on the day and slot. The building is a genuine piece of industrial architecture and the Gravity Bar view over the city is the real payoff, but this is a brand experience rather than a museum, and if your days are tight it is the one large sight we would drop first. Teeling Distillery in the same neighbourhood and Roe and Co nearby both run shorter, cheaper tours.
Cross the river to Smithfield. The Cobblestone runs traditional music sessions every day, starting mid-afternoon at weekends and early evening midweek; there is no cover charge and you will want to arrive before the seats go. A short walk north is the tenement museum on Henrietta Street, a Georgian street that survived largely intact, where one grand townhouse built for a single family was housing close to a hundred people by the time of the 1911 census. It is guided tour only, pre-booking essential, with tours Wednesday to Sunday roughly hourly through the day and later slots on Thursdays. Take it as the other half of the Georgian story you spent yesterday admiring.
Two ways to finish. On the north quays, EPIC The Irish Emigration Museum charges around EUR 20, with a discount for booking well ahead, and the replica famine ship Jeanie Johnston is moored a short walk along the water. Or go out to Glasnevin Cemetery for the guided history tour, about ninety minutes and worth booking, which takes in Daniel O'Connell's crypt and round tower and the graves of Michael Collins, Charles Stewart Parnell, Constance Markievicz, Maud Gonne and Eamon de Valera. Afterwards, John Kavanagh's, known to everyone as the Gravediggers, sits against the cemetery wall. By the pub's own account the same family has run it since the 1830s. There is no music and no wifi, which is the point.