If you fancy walking/hiking, and you want to discover Ireland's beautiful landscape at your own pace, I suggest taking a look at the surrounding area of Dublin
: there are several way-marked paths and loop trails with fantastic views over the Dublin Bay, or you could explore the Wicklow Uplands, depending on the route you take.
For your third base, I'd recommend Galway, as it's one hour and half by car from Limerick, and two hours and half from Dublin. Also, there are two other good reasons to visit this bustling area: The first one is that County Galway was awarded the European Region of Gastronomy
in 2018 for its high quality food and restaurants. Secondly, because it is close to Connemara
, one of the wildest bits of land on the west coast of Ireland, perfect for trekking
If you need further tips to plan your trip, I'll be glad to help you. :)