Train From Dublin Airport to Limerick

You can take a bus (coach) directly from the airport. It'll be faster and cheaper than going into the city and taking the train. The long-distance coaches are very nice. Here are the options:
https://www.dublinairport.com/to-from-the-airport/by-bus/all-ireland-bus-routes
If you're going to stay a few days in Dublin after the tour (if you can swing it, do), then take the train.

For most of the day you join a Cork bound train at Dublin Heuston, as far as Limerick Junction where you switch trains for a shuttle to Limerick.
It is a slick operation, step off one train onto the other, and the shuttle is held if the Cork train is late.
There are two through evening train, and one evening train where you change at Ballyprophy, not L junction.
Dublin has 2 main stations, make sure the taxi or bus is for Heuston not Connolly. Buses go direct from the airport to Heuston.

Thanks all for the replies, once again you all amaze me, SO much great information. We love trains, so I think we will look at the option to do the 2 trains, I am interested to see just how "slick" an operation it will be too :)

I like trains, too, but coming off a flight from the U.S., my choice would be the bus from the airport. Not quite as comfy as the train, but fewer steps while short on sleep. But I guess if it’s your only opportunity for a train in Ireland and you just want the experience, it’s not a bad choice.