Keep in mind that Killarney is a very touristy town catering to those traveling by bus in large groups, of which there can be many. It's certainly attractive enough with a variety of shops, etc. to see, but I'd give it but one day and one night at most. The one pub there I make the rookie mistake of patronizing on my first trip that draws the bus crowds features nightly group singalongs of the tired, old familiar Irish tunes (Danny Boy, etc.). Lively but, well, shopworn and loud.
If you want to experience some genuine Irish culture up close, then go to pubs with informal "trad" (traditional) music sessions at night, starting around 9:30 pm. You can have a good dinner (pub grub) beforehand, all while scoring a prime table, ready for the music to begin. Look for sandwich boards out front announcing "Traditional Music Tonight" or something similar. Trad is soothing and finger-tapping enjoyable, as well as a great way to meet the locals. Most pubs across Ireland host sessions most every summer evening.
In lieu of Killarney then, spend your time in pleasant, vibrant Dingle town (two or three nights - with numerous options for trad) and on the peninsula, where you'll derive much more from the pre-historic beehive huts, a famine cemetery or two, settlements dotting the far western coast and fantastic scenery. A ferry runs from Dunquin at the peninsula's western end to the Blasket Islands, and another service offers various tours from Dingle town. Check them out here:
I hope your time in Killarney National Park includes the Gap of Dunloe and Moll's Gap, both offering magnificent views and scenery.