Car rental and direction of travel

You really don't want to deal with a rental car in Dublin city - it's more hassle than convenience, and parking can be a royal pain. Better to pick up your car at the airport on your way out of town. The rental car concessions open early out there (5:30 AM in some cases) versus later opening times for outlets in the city. You'll find the best selection of vehicles at the airport too as you shop around for a car big enough for 4 people plus their luggage.
Many here (ourselves included) rely on AutoEurope for rentals in Ireland - they offer competetive pricing and a clear explanation of the various insurance covers ... which can be difficult to sort out when dealing with individual company websites.
Your proposed route looks OK. My only advice would be to slow your pace a bit and not try to "see everything" in such a short amount of time. Ireland is a country best savored at a relaxed pace.

Traffic, avoiding "no drive" zones, combined with driving on the wrong side makes Dublin a tough cookie. I feel pretty comfortable driving in Dublin after five trips. Most car agencies have "in town" depots, so you don't have to rent at the airport, although they are in less than convenient places. Make sure you have a good navigator and map, or better yet a dashboard talking GPS that you pre-load with your objectives. We travel with a Garmin device that I've pre-loaded with each day's route and stops, so it's dead easy. I find the built in GPS options to be far inferior, as you have to figure out how they work, then fiddle with programming, on top of paying though the nose.

My suggestion would be to take a taxi or bus upon arrival to your hotel. On the third day, return to the airport, rent a car, then head to Newgrange, then on your way to Galway. Some will suggest taking the train/bus to Galway and renting the car there, although there will be a drop-off fee involved. If you take a taxi, have them take you to the rental office, which is quite a ways from the airport - otherwise, you have to wait for a shuttle bus and stand in line with your fellow passengers.

Booking lodging in much of Ireland is time critical, and finding two rooms makes it worse. You should be booking now!

We always like to have a car so our luggage can "spread out" a bit. We also have a credit card that covers Ireland for the CDW insurance. They often require written confirmation from the card issuer. Ireland car renting is a bit of an ordeal.

Your flight will probably arrive in the morning (especially if coming from North America). You'll be groggy and wiped out on your arrival day, so wandering around Dublin would probably waste that day. Driving on that day, in that condition AND switching to wrong-side driving....just to park the car and pay for it to sit there? Not a very efficient use of your time, money or arrival day.

Consider an alternative something like this:

Skip the car, skip the zombie-day killing time until you can hit the sack, skip the serious risk of a major traffic accident, skip Dublin (for now). Push on to Galway straight from the airport in comfort and with no stress - and let someone else do the driving (someone who's in much better shape for that than you will be). Take the bus ("coach") from just a few steps outside the terminal doors, across Ireland to Galway. In Galway, make your way to your accommodations, eat dinner, get a good night's sleep (or two) before you get behind the wheel and deal with the challenges of driving. The bus is cheap, comfortable, warm, cozy, even has wifi, and would allow you to snooze a bit if you want to. It's super-easy. After you wake up refreshed in Galway (the next day or after spending some time in Galway if you want), pick up your rental car there, and drive on.

At the end of your trip, return to Dublin (maybe drop the car off somewhere else and take the train for the last stretch to Dublin) and spend your last day or days there. It's more efficient to "do" Dublin once, rather than multiple times. If you are flying out of Dublin, you should spend your last night there anyway.

We went counter-clockwise (as described above), like on this map and it worked out well for us:
✹ Flew in to Dublin
✹ Took the bus from the airport to Galway
✹ We flew out to Inis Mor (in the Arran Islands) for a couple days before flying back to Galway
✹ Picked up our rental car in Galway when we actually needed it
✹ Drive counter-clockwise (or "anti-clockwise" as they might say over there) south to Dingle, around the beautiful southwest
✹ Spent out last night with the car in Kenmare, drove to the nearest town with both a car rental agency (to drop the car )and a railway station (to get us to Dublin), then we took the train back to Dublin, where we spent a day before flying home

Thank you for your replies!
Robert: Sounds like we will definitely be better off picking a car up from the airport on our way out of the city. I’ll look in to AutoEurope.
Jjgurley: My son is in charge of navigation so I will pass that info onto him. I’ve added Newgrange to our list of places to stop at. We will definitely get onto booking accommodation this week! I think my husband’s credit card may cover the car insurance but I’ll definitely look into it.
David: We arrive just after lunch in a flight from London. Your map and itinerary are very helpful!

I'd pick the car up at the airport on your way out of Dublin. We've driven a few times in the city and it's crazy town. So much traffic! Honestly, it takes so much time to drive around it's just not worth the hassel as you will not save yourself any time.

Getting to/from Dublin airport to Dublin center is super easy. There's a shuttle bus (Dublin Express) which takes about 20-25 mins and it between 8-10euros one way. Enjoy your trip!

Thanks for the information about the airport shuttle bus KimberlySEA. Sounds like a good way to get into the city