You can be in the center of Dublin in just over 30 minutes from the airport by taking the AirLink Bus. However, a taxi can often be quicker, depending on traffic (taking anywhere between 20 and 50 minutes), and cost you between 20 and 30 euros.
Better than a taxi is booking a private arrival transfer and a private departure transfer so you know exactly how much you'll be spending beforehand.

You can then walk from one end of the city to the other, but if you feel you need to use public transportation, choose the LUAS, the city's sleek light-rail tram system.

For trips to the rest of Ireland, check for the cheapest car rental rates here.