Airport transportation

The best way will always be via private driver or taxi. Door to door.

Six services from the airport call at Merrion Square- 30 minute journey time, with a combined frequency of 8 buses an hour-
Aircoach #701- twice an hour;
Wexford Bus #740, 740A and 740X- each once an hour
Dublin Express #784- twice an hour
Bus Eireann #2- hourly

Private driver is a great way to travel. For the rest of us, there are buses departing from the airport into the city all the time. I wasn't able to see anything direct to Merrion Square, but I have found that Google Maps is excellent for public transportation connections. Here is the Transport for Ireland journey planner for getting to your destination from the airport:
https://journeyplanner-production.transportforireland.ie/journey-planner?origin=%7B%22name%22:%22Dublin%20Airport,%20Dublin%20Airport%22,%22id%22:%228240000005%22,%22shortCode%22:%22%22,%22coordinate%22:%7B%22latitude%22:53.4265700632111,%22longitude%22:-6.23880001666025%7D,%22type%22:%22BUS_STOP%22,%22eastingnorthing%22:%2231712127,24324962%22%7D

With luggage, and/or for more than one person, it's a taxi.

Of course taxi is also the easiest, and door to door without luggage hassles but the express bus (Dublin Express or Aircoach is also quick and easy and inexpensive. Check the stops that stop near your destination,(can't remember now which bus we took) but ours dropped us off near St Stephens Square, so a short walk to Merrion Square.

FYI - idea of costs:
We took a taxi last week from The Chancery (near castle) to the airport at 8 am on a Thursday. The hotel arranged the taxi, four adults, 4 carry on bags. 42euros.

I used air coach to Merrion Square. My hotel was just across the street from the Oscar Wilde house. The bus stopped in front of the Oscar Wilde House, so it was perfect and only €10.