It’s quite easy to see a lot of things without a car if you are going to base yourself in Dublin and Galway. There are a wide variety of guided day trips from both cities so you won’t have any difficulty getting to the Cliffs of Moher. From Galway you can also find day trips to the Aran islands, Connemara region, and others. From Dublin you can get a day trip to Glendalough, Newgrange, or even just go on your own to either Belfast or Kilkenny. Many day trips will make more than one stop, so they are a good way to see a number of sights without a car. This question has been asked several times so would suggest doing a search and reading through the responses to see what may appeal to you.