First Time Visit!

I'm going to nominate Galway because a) I love it!, b) it is a lively city and c) there are several day trips you can do (although some would need to be on a tour bus but this would be the case with most places - you can't do Killarney National Park, the Ring of Kerry, Dingle Penninsula, Beara Penninsula by train).

Trains are quite poor in Ireland, they are good for travel between cities and large towns but beyond that don't provide much.

In Galway you can visit Aran Islands, Cliffs of Moher (probably) and Doolin by bus on your own initiative and Connemarra and the Burren on a tour bus. Perhaps after a handful of days in Galway you head to Killarney for the national park and a coach tour of Dingle Pen and/or RoK and a public bus to KIllarney National Park (the park is huge so that might depend on your mother's level of fitness.

So perhaps Galway two nights, Tip one or two nights and Killarney two nights, getting to and from these places by train (although bus would be cheaper).

are NOT interested in driving; we have driven in different cities in Europe before, but would prefer not to deal with the stress this time around
I would actually advice you reconsider not hiring a car, as it makes it much easier to explore the countryside. There's no need to get stressed about driving in cities, you could easily pick up the car at the airport on your last day in Dublin and return it there before you fly out.
I haven't driven in central Dublin and wouldn't do so unless I really had to, but the other towns I've been to (Kilkenny, Killarney, Tralee, Ennis, Galway and Drogheda, plus a number of smaller places we passed through) were all quite easy to navigate and small enough to make it possible to find a B&B just outside the center, leave your car there and walk around town.

I'd second #2 re: driving.

There are direct buses from Dublin Airport to places like Galway and other major regional cities. There's something to be said for using such a service and hiring a car locally in Galway (or wherever). Even just getting from Dublin Airport to a main road to leave Dublin can be an unpleasant driving experience.

Thanks for all the advice. As I conduct more research, it's becoming very apparent that traveling around Ireland without a car may not be feasible. This is troubling, as we really don't want to try our hand at driving on the left side of the road!

If we stayed in one particular city and then took trains or buses out on day trips to popular areas/attractions, would the odds be good that there would be a tour bus nearby which we could hop aboard? Personally, I would be happy to travel into different cities, walk extensively about the country side, see what attractions I could get myself to, and then head to a b&b for the night, but this trip is important to my mother and I think she will want to see more touristy things. I don't want to disappoint, and so must try to plan as much as possible.

I really appreciate any and all advice on the matter. I'm used to throwing on a backpack and traveling on a whim!

Do you recommend potentially getting a car in the city of our choice, so long as it's not Dublin? What about driving from city to city? If we stayed in Galway and wanted to drive to Tipperary, for instance, would that be a harrowing experience for someone who is used to driving in cities (I'm from NYC), but always on the opposite side of the road?

If you fly into Shannon airport and rent a car, it really wouldn't be too bad. There is no big city at Shannon, so you're not thrown into lots of confusing traffic, etc. I think driving on the left comes fairly easily and if you want to see the Irish countryside, there is a relatively small amount of city driving. Most towns and villages are small.

It is feasible to see without a car, though. I would also recommend Galway as you can do several things by day tour. If you can get to Killarney National Park, that would be another option with lots to see sans car. With only 6 days and no car, I'd recommend only one base with day trips. WIth a car you could do more.

With a car you can certainly do and see more. The train is pretty useless for travel to small towns in Ireland, bus is a much better option but takes a lot longer to get from a to b than if you were driving.

I would recommend a bus tour, all of which go on a clearly defined touristy route. The only company I know is Paddywagon though and that mightn't be suitable for your mum, it's more a backpacker crowd.

Where in Tipperary do you want to see as outside of the main towns it's off the tourist route.