What would a boy's (14 yrs) Sunday Best suit be made of (Eastertime)?
It would depend on the wealth of the family, but probably wool; linen at the most if they could really afford it. If the family is Catholic, it would have been a hand-me-down from an elder sibling.
Would horses still be used for transportation, especially of goods? What kind of horses? I want a big cold-blood breed; they're called Shires in England, but what about Ireland, do they have their own cold-blood breeds?
Cars didn't start becoming common until the 1960s--Ireland was incredibly poor during the 1950s and there was mass emigration. The only people who would have had a car would have been the doctor and anyone from an incredibly wealthy family. Horse and carts were a common sight--there are pictures of my parents riding on them. I don't know exact breeds that are popular here, but the ones on farms don't look that distinctive. The main thing would have been that they stayed vertical for as long as possible.
What about people, how did they mostly get around, if they were poor?
Walked or rode bicycles. It was normal to walk up to two hours back and forth to get to school, according to my dad. (We had many 'back in my day' debates when I was growing up.) There's a book called TO SCHOOL THROUGH THE FIELDS by Alice Taylor which documents the kind of lifestyle I think you're researching.
What about schools? Would children in villages be picked up in buses to go to a central school, or would each village have its own school?
Every village definitely wouldn't have its own school--there are villages near the town I grew up (I currently live in Dublin, but I was born and raised in a town in North Co. Cork) that literally consist of a couple of streets with surrounding houses. It would be common for kids to walk to the nearest school in another town. The Irish government didn't start making compulsory education attractive until the '50s again, and even then both my parents left school before they were sixteen for various reasons (my dad did because, as something like the third out of six or seven siblings, he was expected to get a job to keep the family going instead of continuing with this education nonsense. He still regrets it). If you lived in a very remote area with no way to safely walk, you could get a bus but I'm not sure if you paid for it or not.
What about water supplies? Toilets? In a country setting, would there be indoor or outdoor toilets? Slop buckets?
My dad, again for reasons I'm not going to get into here, was finally able to put a toilet in my grandmother's house... less than ten years ago. It depended again on the family--many houses of poorer Irish people never had toilets in them, and it was expensive to put in all the necessary plumbing. For people who could afford it, toilets and sinks started appearing at the turn of the 20th century, but this was just over fifty years after the Famine--there wasn't much wealth going around. Some people might have had toilets outside, others would have used a slop bucket outside. It depends on the background you've created for your character. I just made sure to use the bathroom if I needed to before we visited my grandmother because I was a bewildered little girl used to having indoor plumbing.
I hope this helps! I know you've said you've chosen a location but I'd also like to recommend
Kenmare in Co. Kerry. Tiny little village in the south-west, would have had a lot of sheep farming, close enough to a couple of bigger towns (which would now be considered cities). There should be some stuff lurking in Google.