A good home base while in the country for 14 days

There aren’t a lot of freeways so it takes a while to travel from place to place. If you’re planning to drive, you’re better off with maybe 3 home bases to see the areas you are most interested in. Consider flying into Shannon and out of Dublin (or vice versa) to avoid backtracking.

Agree with the advice above to have several different bases. I'd probably base in Dublin for that city and a couple of day trips (i.e. Glendalough, etc.). Galway is an easy and fairly quick drive across the island and would be a good base to see places on the west side (i.e. Aran Islands, etc.). Then you can go either north or south for a third base depending on your interests.

I agree that with the time you have available you can probably base yourself in 3 different locations and explore further afield from there.
My suggestion would be to catch the express bus to Galway upon arrival at Dublin Airport. It's a particular favorite here since it allows you to hit the road running, and renting a car there is pretty straightforward if you choose to do so.. There are a number of day trip providers based in Galway that will allow you to experience the charms of Connemara without having to rent a car, though having one is a great way to see the countryside at your own pace.
You could then take the train or bus back into Dublin (it's only a couple of hours) and then continue up to Belfast for several days. There's lots to see and do in the city, and it's a handy base for day trips up to the Giant's Causeway, Derry, and maybe the Antrim coast ... again without necessarily having to rent a car.
Could then finish up in Dublin where there are day trip and tour providers to take you up to Newgrange and the Boyne Valley or south to the Wicklow mountains, Powerscourt and Glendalough.
Having a car would be preferred, but you can put together a very nice itinerary using public transportation and day trips.