We've been to Ireland four times, and have always done it with a car. We live in an area with near zero public transportation, so I'm biased. You will find folks here with positive results using public transport, but:
1) Getting off a bus in an obscure area to walk to a remote ruin or hike leaves you with luggage issues unless you always return to a base, which means you spend twice as much travel time without actually seeing much.
2) We find most interesting places are places we find while enroute to the next town. There may be a bus and it may stop there, but luggage and time are both issues.
3) The train system is very Dublin-centric, meaning traveling from the southwest to the northwest may involve a return to Dublin.
If you're spending five days each in Dublin, Belfast, and Galway the public transport approach is fine. You'll, unfortunately, miss many interesting places although there's lots to do in those towns.
In Dublin, you can do a daytrip to Newgrange and the Boyne Valley.
In Belfast, you can do a Giant's Causeway daytrip.
In Galway you can do a Aran Islands daytrip (or spend the night)