6 nights in Ireland

Staying in Dublin would allow you to enjoy the city and enjoy day trips and tours to attractions close by: Glendalough, Powerscourt and the Wicklow mtns to the south and Newgrange, the Boyne Valley and Tara to the north ... all less than hour from the city.
Other possible excursions, either as day trips or overnighters, would be Kilkenny (about 1.5 hrs by train) and Belfast (2 hrs). Though the Titanic Museum in Belfast is the main attraction, the city itself is beautiful and is worth at least one of your days. There are day tours from there up to Derry and the Causeway Coast - would make for a very pleasant day out on a sunny day, not to mention the history lesson involved.
For something a bit further afield you might consider taking the bus or train to Galway (about 2.5 hrs) and basing yourselves there for a few nights in order to enjoy beautiful Connemara ... with maybe a day trip by ferry over to Inis Mor added to the mix.
There are a lot of options available to you that don't require a car.

Robert gives good advice. Staying in Dublin would make the most of your time. Remember, when you change locations you have to factor in the time it takes to pack up, check out, get to the train or bus, get to you next destination, check in and then find your bearings. You can easily spend half a day changing locations and then you have to do it again to return to Dublin to fly home. All of this time could be better spent taking a day tour out of Dublin to some of the places Robert mentions. If you are relying on day tours to see some of the sights keep in mind that a lot of them are all day, and tend to leave around 9am and return late afternoon. So, for example, if you take the train from Dublin to Galway and you can't arrive before the day tours leave you will be spending the day in town unless you can arrange for a private driver. You say you have six nights, but that means you really only have five full days. That means you would really only have two full days in each place if you were to do Dublin and Galway because you have to allow for travel time there and back.

Thank you so much Robert and Anita! I so appreciate the responses you gave. All really good advice. I have a few more questions that I'll compose a bit later today but I wanted to say a quick thank you now.

My faves in Dublin were (in no particular order) the Dublin Literary Pub Crawl, the National Archaeology Museum, Dublinia (hands-on museum all about the real Vikings), and nearly every pub. I would probably list Kilmainham Gaol but tickets sold out just minutes before I got there. If you like plays, see what's at the Abbey and the Gate - London-quality at bargain prices.

I thought Galway was touristy, but I was there on a weekend. Loved Cobh. Too many places, too little time. The best place to hang with locals is the pub in a small town. But you don't need a pub, everyone has the gift of gab. Start a conversation with every local you meet!

Galway is the opposite of what you described. Extremely “touristy”.

I would Consider 2 nights in Belfast. (Premier Inn Titanic Area was excellent) See the Titanic Museum, walk around etc.
Next Day
(Option 1) take the train to Coleraine. Walk directly through the train station and get on the waiting bus that will take you along the Causeway Coast, including the Giant’s Causeway and Dunluce. A LEAP day card can be a savings here.
(Option 2) sign up for a day tour from Belfast and use their transport.

Consider going directly from the airport to Belfast on your first night and stay 3 nights there and then 3 nights in Dublin. Do Dublin all at once at the end of your trip.

Thank you all so much for the advice! Now Northern Ireland is on the list for sure. Galway has a big festival in July so probably best to avoid it (we're not into too many crowds). We'll just have to go back at another time of year to see the west coast. Now for suggestions on lodging in Dublin and Belfast. Any thoughts? I'll peruse the forum and Rick's ideas as well.

I already mentioned Premier Inn Titanic Quarter in Belfast, but I will mention it again with more detail. Reasonably priced hotel just a few moments walk from the Titanic Museum with an excellent location on the river. About 10-15 minutes walk along the river to downtown or catch the bus outside the door.

Another vote for the Premier Inn in the Titanic Quarter. Nice hotel, great location, resonable prices, and the HoHo bus stop is right outside the door. Hard to beat.
Try to be at the museum when they open in order to beat the crowds - it's a very popular attraction and fills up in a hurry once the day trippers and tour buses start to arrive around 10 AM.

Great, thanks for the suggestion!

Regarding your search for Dublin accommodation: the closer you look near Trinity College and the Temple Bar area the more expensive it will be - prices generally decrease the further away from the city center you go. The tradeoff will be convenience versus cost - the money saved by basing yourselves a little futher afield will be offset by the cost of the taxi or DART to get to the collection point for your tour.
I'd avoid the Temple Bar area as a general rule. The evening revelers raise a pretty good racket until the wee hours, and since most places don't have A/C you'll have the windows open, making it sometimes difficult to get any sleep.
We've stayed at this place a couple of times and can recommend it: https://www.leesonbridgehouse.ie/. It's a half mile level walk to St. Stephens Green, and the stop for the Airlink bus to/from the airport is right outside the door. Very convenient. Good breakfast too.

Thanks everyone for all the great suggestions! We have a few decisions to make now. If I have any more questions I know where to turn. :)

If youre in Dublin check out the Chester Beatty Library. Its the best museum in the city and no one seems to know about it.

Thank you!

You may find a convenient base in Dublin at one of the StayCity managed apartment facilities.
Fully equipped with services available, and easy walk to city sights. I am reminded of a stay there across the street at StayCity Christchurch.
Very convenient to Trinity College, museums, TBar area, etc.
With kitchen, laundry and parking garage nearby, we found it secure and comfortable family base.
Cheers and Safe Travels!

Thank you Pete!