First Time in Ireland and Concert in Cairde na Cruite

Do you plan to rent a car?

Yes, we will rent a car. Advice on that topic is also welcomed. Thank you.

After a day or two in Dublin, suggest collecting your car at the airport as you head out of town in order to avoid the hassle of driving in the city. The airport vendors have the largest selection of cars to choose from, and you'll have a better chance of getting an automatic, which I highly recommend if you're new to LH driving (the default rental in Ireland is a stick shift, fyi).
With 9 days on the ground you can organize a nice loop drive from Dublin - maybe heading west to the Galway area first since your concert is a hard commitment on the 1st. Though Galway town isn't one of our favorites, the Connemara area is. Basing yourselves outside Galway (in Clifden or Westport for example) would make for easy explorations of the prettiest part of Ireland, including a day trip over to Inishmore via the ferry at Rossaveal. From there you could continue clockwise up towards Derry and the Causeway Coast. A good base for the area is Portrush.
You could then finish up in or near Drogheda, which is convenient for your event at Cairde na Cruite as well as Newgrange, the Boyne Valley and Tara. From there it's only 30 minutes or so to the airport to return your car.
Driving your rental car from the Republic into Northern Ireland is pretty seamless, but do check with your car rental company to ensure that there are no surprises re your insurance once you cross the border. Many here recommend Dan Dooley for rentals - they offer competitive pricing and are one of the few companies that displays the insurance details right up front. With most of the others you have to drill down into the fine print to verify what's covered and what isn't (and the cost).

Much sincere gratitude for your kind reply. Alternatively, if we do a mulit-city plan and fly into Shannon and out of Dublin how might our itinerary look?

Many sincere thanks.

Would streamline your itinerary a bit to fly into Shannon, collect your car there, and drive up to the Galway area.
Picking up the car in Shannon would necessitate a one-way rental, but that's usually no big deal since both the pickup and dropoff points are in the Republic. Would be a different matter if you were picking up or dropping off in NI which would make it a one-way international rental, thus entailing a steep surcharge. The one-way drop within the Republic (ie Shannon to Dublin) will be minimal.
By doing it that way you could save Dublin for your last couple of days. Recommend dropping the car at the airport on your way south from Drogheda and taking the Airlink bus or a taxi to your accommodation in the city. Having a car in Dublin is a hassle - more of an impediment due to traffic congestion, unfamiliar rules of the road, and parking issues.
Note that NI uses the pound and that the traffic signs will be in MPH versus KPH. Local currency can be had by visiting any ATM, though most places you'll visit in the North will be happy to take credit cards. Other than that you'll hardly notice any difference while driving.

Thank you for your kind reply.