We spent a week in Kenmare last August and loved it! The kids had a ball as we found so much to do. There are many lovely walks all around the Kenmare area but there is also a buggy/toddler friendly walk in Kenmare town itself. It winds through a leafy wooded area and then runs alongside the water. It is lovely if you get a good day. You can even do some bat spotting!
We did a lot of driving with our 2 children (4 and 3 at the time) to get to places that we specifically wanted to visit such as Dingle to see the dolphins but if you didn't want to travel so far there is a lot to choose from close by. The best thing to do is visit the Kenmare tourist office in the town centre. They have a host of leaflets and were really helpful.
The Killarney National Park is stunning and we drove through this most days. There are so many scenic viewpoints not to be missed and well worth a photo. Definately take a hike with the kids up to Torc Waterfall. The walk is a loop back to your car and our kids managed it no problem.
Muckross House is stunning. We didn't get inside the house itself as the grounds were enough for us. Would recommend hiring a jaunting car or horse and trap that takes you to the Lake or the Abbey (whichever you choose).
As for eating out; well we would highly recommend Mulcahys. We were not sure if it would be child friendly but it was excellant. Lovely atmosphere and fantastic food. Only go in if your kids can sit still through a meal though! Bring plenty of crayons and colouring books!!
We were also told Packies (next to Mulcahys) and Leath Phining Eile are very good but didn't visit these so not sure if child friendly. The Landsdowne Hotel is child friendly and the food is ok but nothing special.
Davitts and Coachmans are good pubs. May need to book a table as can get very busy. Great music in Coachmans at night time if you are going out alone. Crowleys bar is a must for a pint at about 9pm!!?? (so I have been told).
A treat is to visit Sheen Falls Hotel for a club sandwich and a glass of Chablis for lunch. Not sure if this will be open over the New Year though.
If you are looking for a beach (In winter!) then Caherdaniel is stunning. You never know, you could get mild weather and go crabbing! The kids loved this beach. It's a bit of a drive but worth it.
There are numerous parks and sheep farms in the area so check them out as well.
If the roads are not icy, its a clear day and you are in an adventurous mood, drive the Gap of Dunloe. Stunning, stunning, stunning scenery. Take your camera and also be prepared to get a little lost on the way.
Have fun!