Hi Kirsty,
Thanks for getting back!
The train company in the Republic of Ireland is
Irish Rail. You will be able to book tickets and see the timetables on their website. They operate a rolling 3-month timetable so the dates in July you are with us may not be available to buy online yet but keep looking!
Bus Éireann operates the national bus network, so they are great for trips as well. In
Dublin, there are plenty of
public transport options and tour operator companies that offer day trips.
Paddywagon Tours,
Wild Rover Tours, and
DoDublin Tours would be examples of the many operators to choose from.
Belfast has plenty to see and do too. There is a
hop-on hop-off bus that you can use to get around the city and beyond. That said, most attractions are within walking distance in the city and it is a lovely place to walk around.
I hope this helps and you both have a great time with us. Feel free to ask any more questions and let us know when you are here and what you have managed to see.
Warm regards,
Martin