First stop, read about Hiberno-English. It's the dialect that is spoken across the island, although Ulster English has some very unique phrases in itself. I'm not from Northern Ireland (I'm from Galway in the West but to a Northerner I'm in the South, not the south - The South.
but in general the following advice will be fairly universal.
Vocabulary
As you might expect, we use many British expressions rather than the American ones, but we don't think of them as being exclusively British, they're just as Irish to us as our unique expressions.
Americans are
Yanks (all of you, not just New Yorkers!!) and we're Paddies. Like most things Irish, it's
informal.
Apartment or Flat - Both! In a smaller rural area, Flat will be far more common. Apartment is more formal and used while searching for somewhere to live. Informally you may live in a Flat, Digs or Gaff. (Gaff is common in Galway not sure about NI).
Lift is the standard term. Daycare is Childcare and the Childcare centre is a
Creche (French term), Jobs are still Jobs and cars are still cars.
We say
fridge, (never refrigerator)
A
press is both cupboard and closet
A pair of
trousers is a pair of pants. Pants is increasingly used.
Brilliant instead of Awesome.
A
rubber is an eraser not a condom!!
Men do not have
fannies! Only women do and it's a very rude term here!
Arse not ass. Ass is a type of mule/donkey.
Runners instead of sneaker/trainers.
Fag is a ciggarette, not a homophobic slur.
Yoke instead of thingamabob
This is a useful site (our version of urban dictionary) but it seem to lean towards Southern Slang and not Northern. However, at the bottom of the page, you can choose which regions slang you wish to view.
http://www.slang.ie/mostcommon.php
And finally, we speak very informally most of the time. It was quite startling to me when I went to the US, and everyone kept calling me 'Sir'. It was unusual and feels a little uptight. I'm sure I offended people by not referring to them as 'sir'. You almost always refer to your boss by his first name here, even if you just met for the first time but not clients unless they ask you to. But we still don't use Sir or M'am for clients. Too formal/awkward.
Grammar.
Forget about using the Past Perfect in speech. We don't properly use it at all. Instead of saying;
I have done X I have finished Y
We say;
I'm after doing X I'm after finishing Y
The plural form of 'you' is 'ye'.
Go to youtube, and search for some video bloggers from Northern Ireland. It's the best way to understand how people talk and use these words/phrases.