I know there is daily bus service because I met someone who had just arrived from Killarny. Ireland isn't well known for an efficient bus system and she was upset about the delays encountered along her journey. She arrived at 9:00PM and the scheduled arrival in Dingle was supposed to have been at 6:30PM.
So at least one bus a day from Killarney, with a scheduled arrival of 6:30PM.
The town of Dingle is small enough that it would be easy to get by without a car. Some great hikes up into the hills as well. But you would miss a lot of the peninsula if you were on foot.
Dingle Town and Dingle peninsula are unfortunately out of the train-system in Ireland. The closest train-station is in Tralee and from there you may take the public Eirean bus #275 that will take you directely to the main street of Dingle - town.
To explore Dngle -peninsula it is best to rent a bike ( everything is quite flat except the Connor-pass of course). The fee for renting a bike is around 10 Euros a day :
Click here for renting a bike in Dingle
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For rental-cars in Ireland click here !
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for the schedule of buses click here !
So if you're in Kerry and want to visit Dingle, which is the main town in this area, you can get bus 275 from Tralee. If you're in another part of Kerry, see if you can rent a car, otherwise you'll have to get the bus to Tralee and make the connection from there.