Ireland vacation

I'm not very familiar with Galway, but for Dublin and Cork, I wouldn't stay near the train stations. Dublin Heuston is well connected by bus to the city center. All three cities will have taxi service and likely a taxi rank just outside the station.
On my last trip, we stayed in student accommodation at Trinity College, as we were two single women. I don't know how suitable that is for a couple. This time, we'll be in Zanzibar Locke, which is an apartment hotel near Ha-Penny Bridge. AirBnB has become kind of controversial in Dublin because of the impact on the local housing market. I'm an AirBnB user (and host), but Dublin is a city where I don't use the platform for this reason. In Cork I'll be staying in an apartment I found on Booking.com.
The Forum includes loads of hotel recommendations for all three cities, but especially Dublin. I recommend using a site like booking.com to find a place that suits your needs, whatever they may be, and then try to book directly. Booking has all types of accommodation, including hotels, apartments and traditional B&Bs.

Your itinerary sounds good to me. Assuming I am correct in thinking you are flying in and out of Dublin my only possible suggestion would be to consider taking the bus to Galway upon arrival in Dublin. The bus picks you up right outside the terminal. There's an express bus and also ones that go through Dublin (giving you a little tour), before taking you to Galway. Going to Galway first allows you to consolidate all of your Dublin time at the end of your trip. https://www.citylink.ie/timetables/