Once you arrive at Dublin International Airport, there are several ways to get to the city.

Whichever way you may choose, make sure you buy travel cards/tickets. Here are the best offers which you may find.

One day rambler ticket, valid for all Dublin buses for a day. Remember that only the bus from the aiport to the city centre costs 6 Euro (Airlink). The bus stops right in front of arrivals terminal and it is green in color. It is a direct bus from the airport to the city and operates every 10 minutes. It goes to two main station in the city- Busaras (the main bus station) and Heuston Station (The train station).
The rambler tickets can be purchased for 1, 3 or 5 days. The prices are 6, 13.30 and 21 Euros. This is a great saver if you're planning to use the transport outside the city centre. The city centre fare is only 50c so you won't really save that much.
More information about the fares and the tickets visit the web site of Dublin bus here.
Alternatively you can take bus number 16A which operates normally from the airport to city centre and stops at every bus stop. The fare is 2.20 Euro.

If you're planning to visit many places in Dublin my advice is to get the Dublin Pass. The prices are much higher but this pass is a great value worth over 145 Euros of entrance fees.
When purchasing the Dublin Pass you get free admission to 31 Dublin attractions:
Airfield Trust, Chester Beatty Library, Christ Church Cathedral, Croke Park Experience, Dalkey Castle & Heritage Centre, Dublin Castle, Dublin Writers Museum, Dublin Zoo, Dublinia and the Viking World, Dublin's City Hall, Glasnevin Cemetery, Guinness Storehouse, Irish Museum of Modern Art, James Joyce Centre, James Joyce Museum, Kilmainham Gaol, Malahide Castle, National Aquatic Centre, National Gallery of Ireland, National Museum of Ireland, » National Wax Museum Plus, Number Twenty Nine - Georgian House Museum, Saint Patrick's Cathedral, Skerries Mills, Tara's Palace,  The Casino, Marino, The Fry Model Railway, The Old Jameson Distillery, The Shaw Birthplace.More information about the attractions here. Information about the prices.