Killarney is on the Tralee-Dublin train line. It is about 4 hours to get to Dublin from there.
If you want to go to Limerick or Cork by train, you will have to change trains at either Limerick Junction or Millstreet.
To go to any other town, its best to either rent a car or get the bus
You can get to Killarney by train or bus if you don't have a car. There is train service between Killarney and Dublin, Limerick/Ennis, and Cork, as well as a link further west to Tralee.
The city itself seemed small enough to walk around. If you're set on seeing the Ring of Kerry, there're plenty of tour buses that leave on a regular basis, or you could rent a car. Another option is to rent bikes or just walk it to wherever the path takes you.