Best place to base for 4 nights to see...

That is a LOT of driving in a 4 day period. A lot depends on how wedded you are to those particular sights. The basic consideration is do you want to see sights or really experience the west of Ireland.

For example....Blarney castle is on Rick Steves top ten tourist traps of Europe. I deeply regret the time I wasted going there on my first rat race trip to Ireland. We wasted so much time on that trip trying to see too much and learned the hard way less is FAR more. Blarney was a low light on an exhausting and not very rewarding race to see "must see" sights. Some folks enjoy Blarney......and that's fine. But honestly I could easily name a hundred more interesting sights that connect to real Irish culture and history....and are just far more fun and interesting.

The Cliffs of Moher are beautiful but also tourist overrun with tour buses, the strip mall at its base and waves of tourists. Despite that it's worth a visit if you have time. But with 4 nights do you have time to spend half the day driving. But if your looking for insanely beautiful cliffs, windswept wild Ireland with hardly a tourist around the Cliffs of Kerry are FAR better and are right there off the Ring of Kerry on the Skellig Ring. No need for many hours of driving...it's right there with views of Skellig Michael, near Portmagee and Valentia island.

The Skellig Ring and the south side of the ROK offer things that could fill weeks.......ring forts, castles, Abbey ruins, Skellig Chocolates, Bray Head, the Glen, some of the best scenery in the world, beaches, horseback riding, great towns like Waterville, Sneem, Caherdaniel, Cahersiveen, Ballinskellig and Kenmare......and all the wonderful things near Killarney (don't stay in Killarney itself - tourist overrun and very hokey leprachauny stuff) but near Killarney like the Gap of Dunloe, Ladies View, the Black Valley, Bally Beamagh Pass, Ross Castle, Muckross House, Kinsale Sheep Farm etc.

There is so much to do. I'd stay in Kenmare with easy access to the sights near Killarney, the amazing south side of the ROK and the Skellig Ring, the stunningly beautiful Beara Penninsula and maybe take a longer trip to Dingle to drive out Slea Head which is incredible. Kenmare and other towns like it have plenty of pubs, trad music, restaurants and shopping and hotels or B&B 's and offer far more of a real Irish experience than being in one of the tourist centers.

Galway is a pretty city.......but a real city......and a long drive from other areas. It really depends on what you personally want. With 4 nights you can't see everything....and having to drive that much to see all those sights listed can be self defeating......So don't try. There is so much to do based in a place like Kenmare it could fill weeks.....4 nights you wouldn't be bored and you would see a ton of diverse things and not spend all the time in a car.

Hope you have a great trip....but don't make our first trip mistake of trying to do too much over too wide an area. Ireland is fantastic....soak in the experience !

I agree that this is a lot ( too much?) to cover in 4 days. What are the top must sees/do for you? What do you love to do/spend your vacation time?

There really isn't a single base from which you can explore the sights listed, and you'd run yourself ragged trying. Driving times in Ireland (especially in the west) are much longer than Google Maps would have you believe. Most here suggest adding about 30% to any drive time estimates you see from any of the mapping apps.
That said, your better option would be to find a place near Galway from which to explore Connemara, or someplace between Killarney and Dingle which would allow easy access to the ROK and the Dingle peninsula. With only four days at your disposal you'll barely scratch the surface in either location. Better that than racing from one destination to another.
Suggest taking a look at the stretch between Castlemaine and Glenbeigh near Killarney. Basing yourself there would allow easy (ie short) day trip excursions to Slea Head Drive & the Dingle peninsula, as well as being proximate to the western side of the ROK (the prettiest part anyway), including the Gap of Dunloe and the very pretty detour onto the Skellig Ring Road with all of its charms as mentioned earlier.
Kenmare would also be a good central location, and the Beara peninsula is terrific. You wouldn't go wrong if you elected to stay there as a base.
In April you'll be ahead of the big tourist crush that begins in June, so Galway might be a possibility if your preference is to explore Connemara which (full disclosure) is our favorite part of Ireland. Basing in or near Galway would permit easy day trips out to the Sky Road just west of Clifden (another good place to stay), the ferry out to Inishmore, and generally meandering around the Connemara Loop to the National Park (nice easy walks), Kylemore Abbey, and down to the village of Cong. There's lots to do in the area, but its charms are much more subtle than the ROK.

And I agree with the assessment that the Cliffs of Moher and Killarney town are easily skippable, although a day trip into Killarney from a base outside the town would allow you to see Muckross House and the other attractions in the area, and then leave the congestion of the place behind you when you depart in the afternoon.
No matter how you decide to organize your trip it will be great, but I too suggest slowing it down and concentrating your available time in one area.

Forgot to mention Castlemaine .... A great option for the ROK and Dingle. We've stayed there for a week and it was wonderful. If you go there try the Anvil pub......it's excellent !!!!