First things first, when you go in a pub in Dublin order a pint of Guinness, the local beer, or more specifically called dry stout. This is a black beer with a distinct burnt flavor, you might not enjoy it so much at first but many learn to like it and really enjoy the taste afterwards.
The best place to drink a glass of Guinness in the whole world is at the Guinness Storehouse, a popular tourist attraction at St. James's Gate Brewery in Dublin, you can take a self-guided tour there that ends on the Gravity Bar where you will probably get a fresh pint of Guinness, straight from the brewery. This is a great way to learn about beer in general and about Arthur Guinness, the creator of this fine beer type.
The famous Irish pubs are in Dublin's cultural quarter, Temple Bar. And there you can enjoy live music and also some great food coming with the beer in pubs like "The Temple Bar" or "The Oliver St John Gogarty Pub".
If you are not really into beer you can try the Irish whiskey, for a great experience cross the river Liffey and go to the Jameson Distillery on Bow Street. Here you can find out the history of the drink and especially the differences between Scottish, Irish and American whiskey.