Short trip...with kiddos

I'm assuming that you plan to have a car. If that's indeed the case then your first challenge will be to find one big enough to handle 5 people plus their luggage - not an easy task. One possible work-around would be to catch the train or bus to Galway upon arrival, settle in to your accommodation (ie dump the bags) and then pick up your car there ... maybe after a good night's sleep the next morning. That way the vehicle only has to service the 5 people for day trips and excursions but without the luggage - a little easier prospect. You'd still be looking at a full sized vehicle but at least you wouldn't require a van to move everybody around.
Since you'll be going to Galway anyway it sounds like that will be the easiest option as a base of exploration. In late April/early May you'll still be ahead of the big tourist influx and so you should be able to find a self-caterng place in or near the city at a reasonable price. Self-catering will allow you to prepare most of your meals in which we've found is a big cost saver, and that's just for the 2 of us. We've had good luck using booking.com as a search engine and can recommend it. We've also used these guys on our own previous trips: https://www.shamrockcottages.co.uk/. Some very nice properties there with room enough for 5 people to spread out and not feel pinched for space. A self-catering cottage or vacation home will likely be your cheapest option for your group as well.
When doing your searches, make sure that you take a look at the weekly rates for the various places. Many of them will require a 7 nite stay as a minimum, but we've generally found that the prices are so attractive that we save money even if we stay only 4 or 5 nights versus a hotel or B&B.
With Galway as a base you'll be able to explore what we think is the prettiest area in Ireland - lots to see and do in Connemara, including the ferry out to Inishmore or the shorter ride out to Inishbofin. The Connemara loop Drive is worth a meandering day, and with the kids you may want to look into the Hawk Walk at Ashford Castle in Cong. It gets rave reviews from everyone on this forum.
Once you finish your time in the area you can return the car and train back into Dublin for your return flight.

I would for sure rent a car. We had a small SUV for 4 of us and it worked well as long as you keep luggage in check. With 5-7 days, are you open to travelling with carry-on?

I would split my time between Dublin and the area around Killarney and with 5 people, would look to book an apartment which will help keep out low as you can cook for yourself. With most B&B and hotels, you would need 2 rooms for 5 people unless you can find accommodations that offer what they call 'family rooms' which are usually 1 double and 2-3 singles.

In Killarney, we really enjoyed the National Park, The Kissane sheep farm for sheering and heading demonstrations. You could dive the Ring of Kerry and make stops along the way.

My kids still talk about Ireland being their favorite family vacation.

Don't forget you'll need booster cushions for both children. Either bring your own or check that the car rental company will provide them for a fee (having seen the state of some of these child seats I wouldn't want to put mine in them!) Depending on the type of seat you may find that fitting in an adult in the back may be an issue in which case you'll need to start looking at a vehicle that carries at least six passengers. Unfortunately those vehicles jump up in price.

THANK YOU ALL!!! It will be the first time going to Ireland with the family. The thought HAD been take bus from DUB airport to Galway. Rent a car (SUV???) & rent a cottage through Shamrock Cottages for a week. Day trips to Aran Islands, Connemara, & Moher.

HOWEVER....there is talk of staying 2 weeks. Should we go somewhere else for 2nd week? My kids are 10 & 7 so ideally one or 2 "home base" locations would be better than a bunch of different places. Also, benefits of cottage rental vs B&B's?

THANK YOU!!!!

A

I would go to Killarney for the 2nd week. I have never been but its on my list for someday.

Advantages of a cottage over b&b?
1. More space
2. More space
3. More space
4. Kitchen for light snacks and/ or breakfast
5. Washer/dryer

Price wise if you stay for a week, we found it to be more economical than a b&b on a per night basis. If you decide to stay somewhere near Killarney for a week I can recommend a beautiful two bedroom cottage in Killorglin. We rented through Sykes Cottages.