As a former resident I have my preferences. In the city I would recommend Trinity College (and the Book of Kells), The National Museum (a real gem), St Patrick’s Cathedral, and for traditional music O’Donoghue’s on Merrion Row. There are many other fine things to see and do in Dublin, but as we used to say, ‘Dublin isn’t Ireland’; try to get out of the city for at least a couple of day trips. South of Dublin in beautiful County Wicklow is Glendalough, an ancient monastic site with a round tower; nearby is Powerscourt with its waterfall and gardens. To the north Newgrange is well worth seeing.
We enjoyed Kilmanham Gaol. I have mostly visited family in Dublin so did not spend as much time on the traditional tourist sights.
http://kilmainhamgaolmuseum.ie
The OPW site provides comprehensive reviews of each of their properties, neatly organized by geography.
I recommend the Dublin Literary Pub Crawl, led by professional actors. It’s an enchanting evening’s entertainment. You hit about a half dozen or so historic pubs frequented by famous Irish writers, interspersed with stories about said writers and readings/performances of scenes, poems, literary passages, etc. It’s all a bit of fluff; it’s not as if you’ll gain some profound new insight into Joyce. But it’s thoroughly engaging and enjoyable. It gives you a bit of an appreciation for and an occasion to reflect on Irish writers’ enormous contributions to English language literature, and it’s an opportunity to knock back a pint or two in a safe setting if you’re so inclined. For those who don’t imbibe, the pub stops would probably be tedious. For the serious scholar of Irish literature, it might be thin gruel. For those who want to delve deeply into the pub scene, the pub stops are probably too short and the literary claptrap just so much noise. But if you enjoy an eclectic mix of literature, theater, history, pub culture, and a casual pint, it’s tough to beat.
The Hop On Hop Off bus is a great way to see the city, as westcoastmomof2 suggested.
Don"t miss Kilmainham Jail. Will need a reservation.
Glasneven Cemetery
Enjoyed Phoenix Park when I was there last.
Loved Powers Court Gardens, a relatively short bus ride from the city.
The Temple Bar District is really cool.
Do the Guiness Brewery tour if only to learn about the process of making beer and the bar and view at the top is enjoyable.
Those who aren’t Guiness fans will find Smithwicks much more drinkable.
Check out St. Stephen’s Green
Trinity University.
On your return trip or if you can add a few days:
Cliffs of Moher
Ring of Kerry
My wife (Dual US and Irish Citizenship) still has over a 100 relatives living there, so we go over every few years.
Busily taking notes for an anticipated trip to Ireland within a couple of years!
@MYOS1634 “I was thinking 3 days in Dublin and 2 Day trips, does that sound reasonable ?”
If you fly to Kerry, it would make sense to stay overnight (on the Dingle Peninsula) and try and do 1.5 or 2 days there. If you want to do 2 day trips, you should do places that are closer to London.
Have fun.
Should have looked up the spelling…
Powerscourt (one word) Gardens
Trinity College (not University)
There are so many interesting larger cities (Galway, Cork, etc.) and counties with delightful towns, villages, and the like, but the most memorable takeaway? The people!
Depends entirely on what route you take. There’s a modern limited-access motorway (the equivalent of a U.S. Interstate Highway or a German Autobahn) that will whisk you between Dublin on the east coast and Galway on the west coast in just over 2 hours. But I agree, the smaller country roads can be very slow. Still, you should be about to drive from Dublin to the Cliffs of Moher on the coast of Clare in about 3.5 hours or so, making it possible to do as a day trip if you’re so inclined. The Cliffs of Moher are truly spectacular. For my money, Galway and Clare are some of the most beautiful parts of Ireland.
If it is raining or you want some thing low -key to do -the House number 29 museum is a Georgian townhouse
The kind of think you would go to on a school field trip -but I liked it. I also liked the Jail.
Happy Bloomsday. 
Thanks!
I like “school trip” type things 
and apparently it rains a lot in July so inside activities will be appreciated!
I second the @Irishdoctor suggestions. (Definitely catch a night of traditional Irish music.)
Belfast and Brey.
Fish, chips, ice cream.
Gardens, zoos, house tours.
Traditional (but non-touristy) craft stores for swag.
The sea/beach.
Galway. Dingle. Aranmore.
A ceili night.
Beef stew made with Guiness, potato leek soup, “the full Irish” breakfast, scones and soda bread, bangers and mash…
If you are a Joyce fan, you can get lemon soap at Sweny’s, which is now operated by volunteers, as a non-profit:
“Today, Sweny’s is maintained by volunteers, dedicated to preserving the shop as it was in Joyce’s time. Here you will find a selection of second-hand books and the perfect location for daily readings of Joyce’s works. Not forgetting, of course, the lemon-scented soap that made the shop famous!”
“Mr Bloom raised a cake to his nostrils. Sweet lemony wax. – I’ll take this one, he said.”
Ulysses, J. Joyce (1922)
St. Kevin’s monastery at Glendalough is also interesting. This takes you into the Irish countryside, and back in history.
You can also see Riverdance in Dublin.
http://riverdance.com/dublin/
In my experience, people tend not to be neutral about Riverdance–they either love it or they really do not like it.
Thank you all for your advice.
I was actually taken to Glendalough to see "Kevin’s kitchen " and hike to the lakes
And I booked tickets for Riverdance for tomorrow ! (€56)
Had soda bread (strange texture ) and Irish stew. Fish and chips at Bershoff on O’Connell - excellent and relatively cheap.
Got a reservation for Kilmainham Gaol for tomorrow!
Thank you for all your suggestions, so far everything 's been terrific.
Personal add ons: Pecan chocolate tart at Queen of tarts on Dame Street, next to Parliament (?) and the brand new exhibit at the gpo.
Go to Kilkenny! Lovely castle, gardens and arts center.
Was going to mention Glendalough and Kilmainham, too, but see you got there.
Giants Causeway in County Antrim (N. Ireland) is a day tour – buses go there all the time. If you have time, we loved Belfast. Did the Black Taxi tour. These folks had a very interesting tour on the political and historical perspectives of the Troubles. http://www.deadcentretours.com/ POur guide grew up during the Troubles and is now a conflict negotiator and works with young people.
CountingDown, We loved Kilkenny! Did a B and B overnight stay there the night before and after our Ring of Kerry trip.
Oops… we were in Killarney!