Traveling with 1 year old and 3 Year old to Italy and Ireland

Would this be their first big trip (intl or otherwise)?
I don't think it really matters where you go as much as what kind of schedule you follow and are able to retain for the trip. But in terms of location, I'd not do something like Pompeii because the cobbles are hard to walk on, but you've already done that anyway. I've seen others say to skip Venice because who wants to worry about a kid falling in a canal for days? So, I'd pick a large, flat-ish city and rent an apartment and settle into your usual routine. They won't really care much or even remember! In winter, cities are best anyway.

Yes, it is the first international travel with them. We have done numerous flights with them but no overseas travel. Do you know any cities that are good for day trips in Italy or Ireland? I was thinking a home base may be best for them instead of moving every few days.

I would definitely not do a lot of moving around -- a week per city would be nice. I'd pick a big city with lots to do versus numerous train or bus rides. I am not a parent but have been kid adjacent enough to say keep it simple!
Dublin and Turin might be nice? Or for warmer temps, maybe Naples or Palermo?

How many days do you have to sleep in Europe?

Hard to make any sort of suggestion without knowing the time available

We are overnight direct from Ind to Dublin and arrive at 8:30am on 11/22 and leave on 12/1 out of Dublin at 2:30 in the afternoon.

Do you already have their passports? Stephen’s Green in Dublin has a very nice playground.

We already have passports as we decided last minute to make last year just a husband/wife trip.

I'm going to preface my comments by saying I have never been to Ireland and have done zero research on Ireland.

I'm assuming you want to spend some quality time in both countries. You don't really have a lot of time to move around in either country. So I'm just going to throw a few things out for you to think about.

How do you want to get around in both countries? Car rental, buses and trains?

Just for the sake of simplicity, I would look to see where you could easily fly between the 2 countries. A quick Google Flights search shows nonstops between Dublin or Cork to a handful of cities in Italy.

Based on that information. I would look for an interesting location in Ireland to stay for a few days on arrival with easy access from Dublin airport and easy access back to Dublin or Cork for a flight to your preferred location in Italy. After a few days in Italy fly back to Dublin at least the day before your flight home.

Ruling out places you've already been (although you could easily return to 1 of them for this trip), there are flights from Cork or Dublin to Venice, Milan and Pisa. From Dublin you can also add Verona, Bologna, Turin and various airports in southern Italy.

Once you have the logistics figured out, the rest will fall in place.

Definitely fly multicity which means fly into perhaps Dublin, low cost carrier to Italy and home from Italy to US all on one ticket not two one way tickets. Have been to both countries but not with little ones so please rely on someone who can help with logistics of airline flight and what seat you need to bring for the little ones.etc. I imagine an apartment for a week in each would be best. best of luck with your plannng.

Forget going to any museums or shows.

tm,
Two cities, max. Cities are best because you don't want to rent a car and manage two car seats. Will you take a stroller of sorts with you or try to rent one there? Baby carrier/backpack? How will you move around in locations? Lots of logistics to consider. Yes to open jaw (different arrival departure cities, if you choose two), not two one-way tickets. Of course, if you need to go both to and from Dublin, then stay in Dublin for the whole time, or one other location and make the second half in Dublin at the last part (for simpler departure). For a second city at that time of year I would choose London (nearby and short plane ride to/from Dublin), lots of things to see indoors, good parks (if weather permits) many day trips (e.g. Windsor castle). Good variety of activities/sights.
My second choice for other city would be Sevilla, but with your short time frame I wouldn't travel that far.

TBH, with just 9 nights on the ground (8 full days to spend), I would just stay in Dublin, or Dublin and one other city in the Republic of Ireland. You save the cost of second flights for four and take the train to it instead. That way you don't waste two whole days moving to and from Dublin.

Weather will be wintry and daylight will be short, so be prepared for that. Remember, all the people in Europe with small children manage to keep them entertained, so try to find out about parks, puppet shows, musical performances that your toddler will enjoy, etc.
Have a wonderful trip!
Yes to AirBandB or an aparthotel type lodging.

Not sure I understand your reply re passports. Do you have the children's passports already as well?

All four of us have passports. All of your replies have been extremely helpful. We have no desire to really go see shows. We live in a cold climate where once winter hits we consider 20*F warm so we are well prepared for the cold. We are thinking two major cities and will for sure be booking travel insurance to help with any kind of delays that may happen flight wise. We did London last year just my husband and I but after talking to a local I don’t feel comfortable with taking my kiddos there. I’d maybe consider more time in Oxford though so that’s something for me to consider! Thank you all!