Taking trains in Ireland

There’s trains from Dublin to Galway and Shannon other wise you will need to rent a car. Ireland is one of the few countries in Europe that public transportation doesn’t work if you want to get outside of the major cities. There is an extensive bus system but it is very slow to travel on and you will spend most of your trip in a bus!

We have taken the train round trip between Dublin and Belfast, and it was great. For Dublin to Galway, it was cheaper and more convenient to take the bus. There were a couple of different nonstop bus options. I think we used www.gobus.ie. For some other areas, including smaller towns, you'd probably want to rent a car.

There is a train system that connects many of the major cities although not as extensive as you might think.

Think of it as a hub system. You take the train to a larger hub then switch to a bus for other areas.

We spent two weeks in Ireland using only public transportation and did just fine. It isn't as convenient as some other places you might have traveled, but the statement that "you must use a car outside the major cities" isn't necessarily true. Look at buses as well as at trains.

You have to plan carefully and do your research for the public transport option. It is not for people who want to be spontaneous because knowing the options and the timetables matter.

We did feel that Northern Ireland was a bit more user friendly on the public transportation options and loved the Antrim Coast.

Look at the Intercity Route Map on the Irish Rail webside that Emma provided to get an overview of the outlying towns that are serviced from the central hub in Dublin. As others have said, the system isn't the most robust in Europe but it'll take you to most of the major towns.
A car would indeed provide a lot more flexibility and freedom of movement versus public transportation, but if you're absolutely averse to driving then there are plenty of day trip providers available from any of the larger cities to answer your needs.

The best source for all train travel advice and tips to save money is www.seat61.com.

I completely agree about the guy in Seat 61. He's got the whole thing dialed in.