Here in Ireland the way it works is,
you can put a child into playschool until they start primary school. We have an ECCE scheme here so you could see if you're entitled to that. Not sure on the details for people moving to Ireland.
Our first year in primary school is called Junior Infants, most start between 5 years and 6 years of age. Tbh I find the later they start the better as it can benefit them through out the years. For eg years ago when mine were small you started junior infants at 4 years, it meant starting secondary school at 11years (they do 8 years in primary). Which was very young for one of mine as they weren't as 'mature' as my other children so I had to keep them back a year, which has A LOT of red tape and you need to proove it would benefit them by keeping them back.
Once they start by 6 years the 'whole system' is happy. Our schools do accept children moving from other schools and other countries and will put them in the class they find suitable for their age and previous education. Our school system has changed a lot over the years, for the better ime. It use to be that the whole class did the same work at the same time, whether they were able for it or not. Now work is set out to the child's ability. For eg one of mine has dyspraxia and dyslexia and they would go to another room with the special education teacher and work on English and maths for their level and ability while the class completed their English and maths. All schools have their own systems but their main priorities are that the children don't feel any different. With so many children with different abilities and different needs, there's always some movement of children from room to room.
I believe that Catholic schools are the most sought after in the UK, here in Ireland we still have Catholic schools, and we also have Educate together and church of Ireland. Our Catholic schools are not religion heavy, if you want your children to opt out of religion you can let the school know and they will not include them and set different work.
Depending on what you want you choose you need to make sure that the school you like is close to where you live. Schools will choose pupils in the immediate area first. You can apply for schools in other areas but you could be left waiting as they must fill places with children from the area first. Most of our primary schools have a secondary attached or within walking distance or short car/bus journey, so that's another thing to consider. For places that have small towns you can have just primary schools and the kids have to travel to another town for a secondary school. We have bus eireann outside of Dublin, they have school buses you can apply for a place on them, you pay for this unless you have a medical card. Then bus eireann also have their regular bus routes that they can use too. You do need to check the transport available for where you will be living as all Counties differ, some have a regular timetable some can have anywhere from an hour between buses. Dublin of course has a much better transport system, they've Dublin bus, LUAS, Dart.
With regards to housing the smaller the town the cheaper the rent. The likes of big counties like Dublin, Galway, Cork etc the rent is higher. You can get a good size house in smaller towns for a cheaper amount. You can have 2 separate towns in the same County for completely different rent amounts. It depends on what's in each and how close to amenities. It's the same in Dublin, depending on the address the rents will differ.
I'd definitely suggest renting short term or if you got a reasonable price for air b n b. You need to see if you like an area and if it suits your needs.
I think the main things are how close you are to amenities, schools, public transport etc there are places that you would need to drive to the nearest shop. You need to plan ahead for when your children are older and is there things for them to do, will they be able to get to school easily enough. You don't want to have to depend on a car, otherwise you can be driving everywhere for a good few years. I've a sister in a small town and if there's bad weather and she can't drive as the roads in the estates are not cleared and dangerous, the nearest bus is an hour walk, Main roads are always cleared of debris and snow etc. Not good if you've kids to get to school or you've too get to work. She has one supermarket supervalu and they're very expensive so she drives 20 minutes to next town to tesco. Then as kids getting older they need lifts everywhere as friends too far away to walk plus you need to consider safety too when walking. When her car breaks down she's basically stranded.
There are jobs here. You would probably have to travel into a main city if you got something in your area of experience. Wfh is not something that really happens here, there's very few jobs like that here. So nearly everyone travels, most who travel to work in say Dublin use public transport, you've bus lanes the majority of the way and motorways in some counties, traffic can be a nightmare everywhere. You would need to find out first where there are companies that you would like/could work for, maybe then base the housing around that and then the schools.
With wrap around care, it again depends on where you live, for eg my sister used a creche for her children, they went there full time until they started primary school, when they went into primary school, my sister would drop them at creche, at 7am, the creche brought them to school and collected them and brought them back to the creche for after school care and my sister collected them in the evening. She was lucky with this creche as many don't offer this and some don't open until 8am, her hours were 8-4 so it was a life saver for her, expensive but was ideal. So that's another thing to consider, if there's a local creche, as not every town has one. What they offer and if it will suit you. You can also check with the school if they do anything. When mine went years ago there was nothing, just basically dropping them off and collecting at home time, I do be hearing from others how their schools are doing a breakfast club or a couple of hours after school, not all schools some are still just school hours, but worth asking.
It probably seems like a heck of a lot to organise, but you're doing everything at once so can seem overwhelming. List out your needs and number them in order of importance and work from there.
Best of luck