Before you go near the records
Talk to your family. It makes no sense to spend days trawling through websites to find out your great-grandmother's surname if someone in the family already knows it. So first, talk to parents, aunts, uncles, cousins, grandparents and find out what they know. Most families have at least one individual who keeps track of the extended network of relatives, and if you can trace her (it usually is a her), you're off to a good start.
Surnames and naming
Don't place any importance on the precise spelling of any of the surnames you're dealing with. Although the spelling matters to us now, before the 20th century extraordinary variations regularly occur in different records - illiteracy was rife, for large numbers Irish was their native language, and most people simply had more important things on their minds.
The amount of information you're dealing with can grow very quickly, especially in the early stages, so it's a good idea to decide at the outset on a way of storing information that makes it easy for you to find things quickly. Most people pick up and put down their family history project as time allows, and the less time you spend hunting for something you know you wrote down somewhere, the easier the search will be. A shoebox with alphabetical index cards for each individual is perfectly fine. So is a loose-leaf binder. There are also some inexpensive software packages and websites that allow you easily to store and retrieve complex family information.
Start from what you know
The only cast-iron rule of family history is that you start from what you know and use it to find out more. It is almost impossible to take a historical family and try to uncover what your connection might be. Instead, think of yourself as a detective, taking each item of information as potential evidence and using it to track down more information that in turn becomes evidence for further research.
Get some idea of the background to your family surnames and have a look at some of the recommended basic guides to tracing family history. Your library's Online Public Access Catalogue (OPAC) will show you relevant material available in Public Libraries. Most county libraries now also have their catalogues available on the Internet. The National Library and National Archives both run walk-in genealogical advisory services, where you can get personal advice on records and research. There are detailed guides to starting out at the John Grenham's Irish Ancestors site and at Seán Murphy's Directory of Irish Genealogy.
What will you find?
Every family history is different, so you can't say what you will find until you start looking. However, as a general rule, the limit for research is the start date of the relevant parish registers. This varies, with records beginning in the late 1700s in Dublin and some of the more prosperous parts of the east of Ireland, but not until the 1840s or 1850s in many places in the west.