Basicly, once you know what kind of rifle or pistol you're applying for a licence for and what it's serial number is (your firearms dealer will tell you what that is, and will probably walk you through the whole procedure as well), you go to your local Garda station. There, you ask for an application form for a firearms licence. The form looks like this:
You simply fill out the above portion of the form, and hand it in. You probably won't have a Pulse ID (that's the Garda's computer system) if you don't already have a licence, so don't worry about filling that in. Don't forget to sign the form!
The Garda who's processing the form (there's usually one or two Gardai per station who are assigned to handle firearms licencing) will want to have a chat with you, to see why you want the firearm, and to generally ensure that you're not a threat to the public! He or she will fill out the rest of the fields on the form, including a recommendation to the Superintendent (if they're not the Superintendent themself) on whether or not you should be granted the licence. It's usually a good idea, therefore, to call the station before you go there to see if that garda is on duty and to let him or her know you're coming in to fill in this form, as a courtesy.
There's a €38 fee for the licence, which is valid for one year and has to be renewed every August 1st (so if you get a licence in, say, May, that licence is only valid until July 31st that year).
Note that if you're getting a licence for target shooting, you may want to request 10,000 rounds under the "Maximum rounds of Ammunition applied for". It does sound enormous, but in reality, it isn't. If you're getting an air rifle or pistol, pellets are sold in tins of 500, which are often sold in sleeves of ten tins. If you're getting a smallbore rifle or pistol, rounds are sold in boxes of 50, which are usually packaged in pairs, then in bricks of five pairs. And if you're getting serious about competing in a few years, you may be buying batch-tested ammunition (where you go to the factory and spend a morning shooting ten or twenty rounds from several different batches of ammunition until you find the best match to your rifle - at which point you buy a minimum of 10,000 rounds of ammunition from that batch!).
So you will actually find that 10,000 round limit useful, if not outright necessary. And, as the default figure supplied if you don't specify one is 100 rounds (you can't even buy that few air rifle pellets!), it's usually a good idea to specify a figure yourself. Obviously, it's a good idea if you're buying this much ammunition to get a lockbox to keep it in - these aren't going to break the bank, and it's an excellent idea from a security viewpoint, as well as a safety viewpoint, especially if there are children at home.
On the lockbox point, note that the Firearms Acts are currently being changed to make it a legal requirement that all firearms licence holders have secure storage for their firearm, and that the Garda must either inspect it or be shown proof that it is satisfactory. That may sound a bit ominous, but all it really means is that you need to buy a gun safe to store the rifle or pistol in; and your local Garda will want to either call out and take a look at it, or you could show him the receipt or a photo of it installed. It doesn't mean you're going to get a knock on the door from the Emergency Response Unit at four in the morning! And, once again, it's not the best of ideas to have a firearm at home if there are children in the house unless it's secured.
There are other fields on that form, by the way (click on the form to see the full page), but the ones above are the only ones you need to fill in. And so long as you're a member of a Gun Club, it's usually a very painless procedure. The licence should take three weeks to get back to you - if it hasn't in that timeframe, you can call the station and check to see if there's a holdup. It's rather rare, however, for it to take that long.