Advice Exploring Ireland by Train: Tips and Routes?

Hello, we will be traveling to Shannon Airport from London (we are from the US but heading to London first) and are considering exploring Ireland by train. We've heard the train system is a great way to see the countryside, but we're not sure where to start. Does anyone have recommendations for must-see stops or scenic routes? Also, any tips on navigating the train system or saving money on tickets would be super helpful. Looking forward to hearing your experiences!