My husband and I will be staying in Dublin city center for a week during the first part of December. I know we will be wanting to do a day trip or two from there. We will not have a car so we will be traveling by coach or train. Any recommendations of must see towns? We are comfortable to do trips on our own and but we would be open to being part of a tour. We realize it will be cold out and we will be prepared for possible inclement weather. We are open to all suggestions as this is our first time here.
Thank you!

I'd suggest a visit to Glendalough, not far south of Dublin. The monastic site is especially beautiful and the website suggests ways to get there without a car:

http://www.glendalough.ie/

The weather in December might not be suitable for scenic tours, and the short day means that you should not use up much time travelling. If you get a clear crisp December day, Glendalough is a good idea.

City visits are less weather-dependent, nor is loss of daylight much of a problem. Kilkenny is worth considering.

Both Glendalough and Kilkenny are excellent places to visit.