The National Gallery is a fabulous Dublin resource.
As a Dubliner, I forgot about the NGI for most of my twenties, dismissing it as a dismal place I was once taken on a school tour, aged 9. And totally wasted it was on my entire class. However, part my current job involves a certain amount of interaction with Irish art and the NGI has taken on a shiny appeal.
The number 1 thing to mention about the NGI is it's entirely free. There's usually one pay to view exhibition but it's often "foreign" stuff, and who wants to see that old rubbish! They are open all year round, and open late on Thursday evenings. They are participating in the upcoming Culture Night, and I note that the current pay per view exhibition of Dutch Old Master Gabriel Metsu will be free that night.
I find the best way to approach a visit to any big museum or gallery, especially if it's free, is to pick only one part at a time. Otherwise, a serious case of museumbutt can set in, and you will need to sit on every available chair for 5 mins each time you see one.
Today I did the 18th/19th century Irish art section. A lot of people might find this dull and, to be fair, there's a lot of pastoral landscapes and still-lifes of dead game to be ignored.
However, it's worth taking the time to view some of the masters of capturing light like Robert Carver, Francis Danby and George Barrett. If you know anything about Impressionism, it's easy to see the French influence on artists like Roderic O'Connor. Make sure you take a detour to see the Caravaggio and The marriage of Aoife & Strongbow by Daniel Maclise, but don't bother with the tiny Van Gogh they bought a couple of years back: I think that money could have been better spent on a pile of Irish art.
I have also recently seen the 20th century art, which features some Laverys (including his huge masterpiece of Hazel in the artist's studio with a fab. purple coat), and the wonderful Harry Clarke drawings for Hans Christian Anderson's fairy tales but sure take your time. It's not going anywhere.
The real danger for me is the excellent cafe (very good soup, great scones) and the wallet-damaging zone of the bookshop. It's a rare day I come out of there without a purchase. It's a great place to find presents.
The National Gallery also holds an extensive range of daytime and weekend courses, tours, lectures and sometimes even music concerts. Check out their website for details of these; most are free. They are child and wheelchair friendly and they also do audio tours of the permanent collection. Entrances are on Clare St and Merrion Square, though the latter will be closed later on this year for renovations of the older part of the gallery.