Athlone, which is in the middle, just two hours (ish) from Dublin. Has a castle and an old pub. Nearby is the Clonmacnoise monastic site.
I disagree that Limerick is "right across from Dublin." Galway is, though I don't necessarily recommend Galway. Loads of people on the forum love it, though. Nor is Limerick worth your time.
So go to Doolin and see the cliffs and hear the music in the pubs. I think you can take the ferry to the Aran Islands from there, as well.
North of Galway is Connemara, which is very much worth a visit. I think folks like to stay in Clifden for this area.
With a car, you can stay anywhere, so use your preferred accommodation site (booking or AirBnB or whatever), set your parameters for size, price, etc. Then zoom the map area wide and see what you find that suits you. Another approach is to use Google Maps and zoom in on towns on your route to see what is there as far as accommodation, pubs, etc. A town you haven't heard of could be a great discovery.