Planning 4 Days in Ireland :) Please Help

How many nights will you be in Ireland, three? Are you arriving in Ireland after spending the night on a trans-Atlantic flight? I'm assuming you'll arrive in the morning since you say you have four days. Time is the essence here.

With no car and with so little time available your options will be pretty limited.
The obvious suggestion is to just stay in Dublin, enjoy the city, and maybe take an organized tour north to Newgrange and the Boyne Valley or south to Glendalough, Powerscourt and the Wicklow mountains. There would also be the option of independent day trips by train to Belfast or Kilkenny.
If your heart is truly set on the West then I'd suggest continuing on to the Kerry Airport immediately upon arrival in Dublin and basing yourselves in Killarney. Though most here advise against using it as a base, there's really no getting around it since it's the hub for day trips and tours around the area ... including up to Dingle for a day.
I'd forget about trying to add Galway to Dingle - the logistics of getting from place to place just don't work.
Galway would be another option to use as a base for the Aran Islands and Connemara- there's an express bus from Dublin airport that would make it pretty easy to do so, and with a minimum of delay.

Hi Mary :)
I'm not sure how to @ you on this so hopefully this message reaches you, but we will have 4 nights in Ireland. Arriving at 11:45AM the first day, 3 full days and then an additional day where we leave at 6:40PM. I'm saying about 4 full days, because I'm counting the first and last days as half days.

Hi from Wisconsin,
I would still repeat Robert's suggestion of staying in Dublin or a suburb on the DART. And the suggestions of day trip north and or south are spot on.

Oh, by the way, the Ring of Kerry became famous because an early RR terminated near to it. I don't recall, but I think the terminus was Killarney, which now has more tourist nicknack shops than residents. This RR made Kerry was one of the first coastal places in 'the west' easily accessible to tourists. The road around the outside of the peninsula has its many views blocked by 15 foot high hedges. Tall tourist buses have solved the hedge issue but filled the road with their size. Tourist have learned this and moved on to Dingle which is much smaller so it gets crowded. Beara peninsula, also small, will be next for tourists. And it is lovely.

Both the River Boyne and Glendalough with the Wiclows offer soft green views of Ireland that are packed with history.

wayne iNWI

I would also agree with staying in Dublin the entire time and doing a couple day trips. You will actually see more this way as opposed to spending half your time in transit between towns. The other easy option without a car would be to take the train to Galway and do a couple guided day trips from there, but you will still be spending some of your time changing locations instead of seeing some sights. Staying out is the best way to maximize your time on the ground actually visiting sights.

Visiting western Ireland (IE) is best from Shannon Airport not Dublin’s Airport. If you’re flying in after a trans-Atlantic flight don’t rent a car on arrival day because you won’t get much sleep on the plane if any at all. Not a good idea when you must drive on the opposite side of the road with the steering wheel on the opposite side too. Furthermore, driving a car you’re not use to brings challenges on top of everything else.
I’ve visited both the Dingle peninsula and the Ring of Kerry and prefer Dingle over the ring. The peninsula is much more dramatic and rugged and not nearly as tiring to drive. The Ring of Kerry is a very long day where you see the same pretty scenery over-and-over-and-over. Furthermore, you need a minimum of two nights for Dingle and two nights for the Ring of Kerry so you don’t have time.
Because you’re flying into Dublin that deserves three nights it will be easier and less hectic to spend all four nights there. This way you can include a day trip to Brú na Bóinne. Go to the visitor center on arrival day and sign up for a half day tour.
With that said and you still want to visit one sight in western IE, spend your first night in Dublin (arrival day). Before leaving home download Rick Steves app and take his Dublin City Walk 1: South Bank. The next morning rent a car and drive to Dingle (4h 30m) and sleep there for two nights. The next day drive the Dingle peninsula. On departure day drive back to Dublin.