I agree, the Cliffs of Moher are a must-see. When I was there last spring it was a little blustery, but the view is fantastic.

Take a drive around the Ring of Kerry, and if you get the chance, head north to Galway, then back to Dublin. Lots of tourist sites, and friendly pubs/delicious food along the way. Surprisingly, some good AirBnB spots too.

In my experience, the tour companies offer some good deals to get you around the island, but the lack of autonomy might be a downside for some tourists.

Make sure you get a solid day or two in Dublin. They have some good museums, and my wife and I agree that we wish we would have scheduled another day to just wander around the city