the right shoes for Ireland trip (late-June-mid-July)

Your kids will be fine in their regular sneakers.

We loved Ireland... but did run into some serious rain when we went in July. My kids brought trainers and tennis shoes - their feet did get wet but they didn't seem to mind. I took a pair of merrill's which were low profile quasi hiking shoes made with goretex and they were perfect - I could slog through just about anything with those shoes and I wore them most of the time.
For the kids, decent trainers (not heavy on the mesh) should be fine - just make sure they bring another pair to wear if one gets soaked. Also, a waterproof jacket is a must, as well as a fleece or sweatshirt that goes underneath. Have a great time!

My husband and I were in Ireland, exactly one year ago. In fact tonight we were reliving our flight a year ago tonight. We were on a 14 day TAUCK tour. We also spent an additional three days in Shannon, before our tour began. It rained and rained, every single solitary day and night. After awhile the weather kind of grows on you. NONE of us complained one bit. We all expected a lot of rain and we got it. The rain didn't interfere with our touring for the most part. It was absolutely beyond gorgeous there. Our tour had us spend two nights in each hotel. We were less impressed with Dublin and Belfast (northern Ireland). Belfast is depressing, due to its history and wars. It rained the most in Dublin and ABSOLUTELY poured in Belfast. It was the worst rain any of the 26 guests had ever been in. You can't believe how beautiful Ireland is. The people are way beyond fantastic. None of us had ever met such warm people. Dublin is just like any big city. If you judge Ireland by Dublin, you won't be as impressed as when you go in southern Ireland. I just can't express how much we adored Ireland. We want to go back there, hopefully, someday. The kids will be fine with their tennis shoes. They will get soaked though. Just make sure you take waterproof jackets and umbrellas. Sometimes when it rains though, there is such strong winds, as in Belfast. So many umbrellas were just discarded on the streets. Mine blew inside out and "died".

We have not walked Ireland, but have done walking holidays in northern England and Scotland. While Gore-tex lined boots work for wet ground, when it pours or you are walking through boggy moors, your feet will get wet. Gaiters only help but do not keep the water out. I have tried putting the gaiters over the rain pants and the rain pants legs over the gaiters to no avail. Your socks and feet will get wet when it pours. The above said, use good walking socks. I prefer Smartwool socks that continue to function even soaking wet. I usually also have spare dry socks on a walk. Stuffing newspaper into the boots helps soak up the moisture once you get in.