Not renting car-staying in hostel. What is the best way to see sights. Are rail tours a good option?
The train system in Ireland isn't an extensive as in other parts of Europe. In some areas, you'll need to take a bus. You don't say where you will be staying, but there are bus daytrips to areas surrounding Dublin, for example.
Michele,
I know that you're arriving in Dublin. Do you intend to take day trips and return to Dublin each night? Or do you plan to roam far enough--for example, all the way to the west coast--so that you'll be sleeping in several different places? Knowing where you want to go will help us with suggestions on how to get there, either by train or bus--or sometimes both.
If I were young and going to Ireland I would take one of the student hostel bus trips. They sound like lots of fun, you can hop on and off and there are lots of young people from around the world on them. Check with Lonely Planet or go on line to STA Travel and see what you can find. I have talked to students who did this and they had a great time.