Tralee is grand, but Killarney is much better. I'd recommend the Castlerosse hotel in Killarney, they do family specials including dinner and breakfast. They do have self catering but I usually go with the 3 night dinner and breakfast option because it's just nice sometimes not to have to cook! They have longer options too. They are on a cycle path right into Killarney town and national park so you can just leave the car and cycle. From the windows in the hotel you can see the deer down by the lakes. Nearby is a horseriding centre if you're interested and you can be an absolute beginner and they go for a gentle walk through the national park. You can rent bikes everywhere in Killarney but it is a bit pricey and if you can bring bikes that's all the better. The last time we were there in April they stored the bikes in the boiler house for us.
The reason I recommend Killarney is that there is so much to do with kids- muckross cycle paths, walking paths, muckross farm, kennedys pet farm, there's another 'wildlife' park with lots of petting animals just out the cork road, and the hotel itself has kids clubs, playground, tennis court and swimming pool. You can rent a boat and go out on the lake, go to an old leper colony, go to the Gap of Dunloe etc. And it's not that far to the beaches in Inch or Banna if the weather is nice. We always intend to go for day trips to Kenmare or Dingle when we are there but there's so much to do we rarely make it.
The hotel itself is grand, it's not five star, but it's comfortable and clean and very child friendly. They'll give you an extra bed in the room if you want or two adjoining rooms, the meals are nice ( its a three course menu with lots of options) and the breakfast is great- everything from cereal, breadrolls, croissants, cheese, fruit, yoghurt and of course, cooked breakfast. You help yourself.
i know I sound like an advert for the place but we go there quite often and always enjoy it. The kids moan when it's time to go home!