Itinerary for a week or so in Ireland

Cost is not likely to be the main deciding factor in the transportation choice. Both are pretty affordable so ease of connecting your destinations is more important. I probably would keep a car the whole time, since there may be stops that you want to make on the way out of Dublin. Or you could take trains between the first and last big cities and rent a car just between Cork and Galway. You also need not return to Dublin if you schedule flights into Dublin and out from Shannon airport.

I highy recommend flying into Dublin and out of Shannon or vice versa to save you from having to come back to Dublin

I would rent a car. It is a lot more flexible.

Lodging wise, I think that you will either have to do 2 rooms or some hotels/B&B had what they referred to as family rooms which were 1 double bed and 3 singles. You may also want to look at apartments through VRBO or Airbnb

1 - I recommend renting a car (outside of Dublin). You'll appreciate the freedom to move and stop at your own pace. We recently travelled throughout the country with our two teenagers, renting a car as we left Dublin. They loved being able to stop at various places on whims. We did take a train from Cork to Cobh for a different experience and that was worthwhile.

2 - I would highly suggest you look at AirBnB rentals. We stayed in some great 2 and 3 bedroom apartments and townhouses in fun areas of Dublin, Cork, Galway, and on the Dingle Peninsula. We all loved the extra space and it was actually cheaper than traditional BnBs for our family of 4. We did stay in one traditional b-and-b, all of us in one room, which was fine, but we definitely preferred having our own space in apartments and houses.

Enjoy the trip!

Good advice from the others. Would only add the following:
With only a week or so at your disposal you really need to narrow down your itinerary to keep from spending most of your holiday in transit going from one place to another. If you're first time visitors I'd suggest crossing Cork off your list - it's just a matter of priorities. As nice as Cork, Kinsale, and the rest of the south coast may be it really doesn't compare with the charms to be found in the west of Ireland ... whether it be Galway and the Connemara area or County Kerry.
Suggest planning to stay in Dublin for 2 or maybe 3 nights to get over your jet lag, explore the city, and maybe take a day trip to Glendalough and the Wicklow Mtns., up to Newgrange and the Boyne Valley, or even the train up to Belfast for a day. Then collect your rental car from the airport - it's a lot easier starting your driving adventure in left-hand traffic from there than it is trying to negotiate Dublin city traffic right after you get behind the wheel. Plus there's a wider selection of cars available from the airport vendors - better chance of getting an automatic which I highly recommend (the default rental in Ireland is a manual), and a vehicle large enough to accommodate your group plus luggage. The larger vehicle you'll require won't be a problem on the motorways or the National (N) roads, but will be challenging once you get off onto some of the more rural roads, all of which are quite narrow and some of them neck down to a single lane in places, requiring you to use a pullout along the shoulder to allow opposing traffic to pass. It's not that big a deal but can recalibrate your startle reflex the first couple of times it happens.
I see from one of your previous posts that you already know about booking.com. You'll save a lot of money for a group your size by using it (or airbnb) to find a holiday home or apartment to use as a base while you explore. We've found that not only do we save on the cost of the room but having the option of cooking a meal or two in saves money too. You might also google Kerry Coastal Cottages and Shamrock cottages - we've had good luck with both. They have some very nice properties, including a few thatched cottages if you'd be interested in something like that.
My suggestion is to pick one area you want to explore, whether it be Galway, Dingle, or the Killarney area, find a centrally located place to use as a base for several days while you explore the surrounding countryside via day trips. Many here like the area between Castlemaine and Glenbeigh in Kerry - it's central to the Dingle peninsula as well as being relatively close to Killarney, the scenic drives exploring the Ring of Kerry, and the very pretty Skellig Ring Road out near Portmagee. Connemara (Galway) would also keep you busy for several days, what with the Connemara loop drive via Clifden, Kylemore Abbey and Cong, a day trip out to the Aran Islands, and just south of Galway are the Cliffs of Moher and the Burren.
As others have opined, doing your trip as an open jaw (flying into Dublin and out of Shannon) would save you a day's travel getting back for your return flight.
Safe travels.

Loads of tips and info. Thank you to all!