Personally I'd take the train from Dublin to Sligo. The roads will be less busy that in Cork, Kerry or Galway in Summer (and accommodation, food etc usually cheaper) and the scenery is stunning too if you go to the right places. There is some amazing cycling routes inland around Sligo, Leitrim and into Fermanagh or go along the coast.
Northwards from Sligo towards Donegal town, around the Glen Peninsula via St. John's Point, Muckross Head and into Glencolumbkille - then north over the Glengesh pass and through Ardara, Dungloe, Bloody Foreland, Horn Head, Melmore Head, Fanad Head, you can't go wrong. There will be no rail service to take you back to Dublin from up here but Bus Eireann will carry a full sized bike on a bus if there is space in the luggage hold. There is technically a €10 surcharge for transporting a bike but most bus drivers don't bother to add it on.
Or else go westward from Sligo along the coast towards Ballina and to Belmullet, Westport, Achill Island, Clifden.
For accommodation, I could recommend the Surf n' Stay hostel in Strandhill (a few miles out of Sligo city and the nest seaside resort in Ireland(IMO)) or Harbour House Hostel in Sligo itself. Kilcommon Lodge in Pullathomas is a beautiful place to stay. The Doey Hostel in Glencolumbkille is utterly weird but beautiful in a quirky way (but old fashioned and basic), as is Tra na Rossan Hostel at Melmore Head. The Valley Hostel on Achill Island is also good (a pub, restaurant and hostel all in the one building - what a great idea
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I have cycled the whole west coast (and south and north coast) in bits and pieces over the past three years or so and have written about it in my blog. Have a look at it if you like and if want more info on anything feel free to ask -
https://theoldbikeshome.wordpress.com/
