Hi Judy
Welcome to Ireland. Even if Ireland is quite small, I would recommend not to add too many miles on your road.
First , Dublin is worth a few days.
Then going from Dublin to Belfast, you could take a tourist bus ( Extreme Ireland or other companies, tickets available in Tourist offices) or take the direct bus or train.
I would recommend to take a direct bus to Belfast, stay in Belfast for 2 nights.
Take the black cabs to have a tour of the Murals, Titanic Museum, walk back from the Titanic into Town, along the River, then Albert Tower, Belfast City hall tour( free), have a drink at the Crown pub, go to the Queen's University and botanical gardens and Ulster Museum. Enjoy!
Then , I would go the West, Londonderry ( half day) and cross the border to Donegal.( plenty buses 1h30mn journey)
then from Donegal to Galway via Sligo by bus
very nice road