I think you have better flexibility renting a car from Dublin airport, rather than taking a train to Galway. FWIW I think it's a great idea staying in Dublin for 3 days because it gives your body time to adjust to the time zones, what road signs look like and you'll get into the car refreshed (not frazzled by a sleepless overnight flight).

A general point: we tend to drive smaller cars here so hire a larger sized car than you think you need.

It's an easy 3hr drive to Galway and more than half of it is now motorway.

You could stay then in Kerry for the third part of your journey.

I don't know if you live in a city/ suburb/ rural but I'd bear that in mind when planning what kind of places to stay in. A lot of US travellers want olde worlde traditional atmosphere when they travel, others might want something more contemporary. I'd consider that when deciding what type of accomodation you plan on on Galway and Kerry.

I'd definitely suggest staying as close to Galway city centre as you can (eg Radisson) so you can enjoy walking to dinner and a pub at night. While Galway attracts thousands of tourists, it's still a real town where people live and work. Killarney (and Bunratty for that matter) are all about tourism. You can't not visit them but you could get a more authentic experience basing yourself somewhere else. You could, for example, stay in Cork and 'do' Killarney from there (ca 1h15 drive). You still get to enjoy the breathtaking scenery during the day but somewhere less blatantly tourist oriented for your evenings.