We did a very similar trip this spring with just a few more days on our trip. We flew into Shannon airport (our flight also ended up being delayed) and we ended up losing a half day in Doolin, since we got in at night. That left us one full day to explore the area. We did the Cliffs of Moher in the morning and a walking tour of the Burren in the afternoon, and a cruise of the Cliffs in the evening. We did not go to any of the Aran Islands. I wish we would have had one more night in Doolin because we would have loved more time at the Cliffs, more time in the Burren, and a visit to the islands.
We then drove from Doolin to Dingle. We came in on Conor's Pass, which was a lot of fun (some people don't like it) but my husband picked up driving quickly and was comfortable with it. We spent some of the afternoon seeing some of the historical sights that weren't on the outer area of the Slea Head drive and then we spent a quiet evening in town. We made a very full day the next day driving the Slea Head drive, doing some hiking on some of the beautiful coastline and stopping in and seeing the sights (there are a LOT in the area). There is a really neat stop, the Blasket Island Center, that I would also recommend.
The next day we went on to spend 3 nights in Kenmare, and on went our trip...So from my trip perspective, I'll try to give you some insight on yours.
I'm guessing that your Day 3 will be a big day of driving and you may end up wanting to make stops along the way? The Cliffs close down at 6pm in March -- you will definitely want to plan to leave early enough in the day so you can spend a few hours at the exhibit and on the Cliffs. We spent probably 2-3 hours at the Cliffs and I wish we would have had more time, personally. If you enjoy hiking and the outdoors, you may feel the same way.
We didn't make it out onto any of the Aran Islands, but I believe there is an an option for when you sail back from the islands for them to sail under the Cliffs. We took a boat out under the Cliffs and really enjoyed it. I would recommend it if you have time. We also really enjoyed a guided hiking tour in the Burren. There are other historical sights in the area that we didn't have time to see that I think we would have enjoyed. Though, if you are going out to the islands, you will not have time for that though with your current itinerary.
When will you fly out of Shannon? It's a bit of a drive from Kenmare to Shannon and I think it would be nicer to be closer to the airport unless it's a really late flight. You will have to have time to get your rental car dropped and then get checked in for your flight. I'm thinking a few different things here: