Anne, you probably won't get to most of these by just using public transit. If this is very important to you, a car rental may be the best way to visit some of these. You could rent a car for a day here and there as you travel around:
CLARE
St. Brigid's Well - Liscannor
Magh Adhair - Quin
Eye Well - Burren
Tooth Well - Burren
Margaret's Well - Ennis
St. Augustine's Well - Kilshanny
CORK
St. Olan's Well - Aghbullogue
Tobrid Well - Millstreet
Ballinspittle
Inchigeela
Sunday's Well and Mary's Well - Walshestown
St. Finbar's Well, Gougane Barra
DONEGAL
Doon Well, Kilmacrennan
KERRY
Well of the Wethers - Ardfert
St. Dahlin's - Ballyheige
St. John's Well - Dingle
St. Erc's Well - Listowel
St. Eoin's Well - Listowel
St. Michael's Well - Ballymore West
Lady Well - Ballyheige
KILDARE
Earl's Well - Kildare
St. Brigid's Well - Kildare
Father Moore's Well - Kildare
MEATH
Tobar Patraic - Ardmulchan
St. John's Well - Warrenstown
Tara (Neamnach, Toberfin, and Leacht) - Castlebye
ROSCOMMON
Tober Oglalla - Tulsk
St. Lassair's Well - Lough Meelagh
St. Attracta's Well - Monasteraden
SLIGO
Tobernault - Sligo
St. Brigid's Well - Cliffony
Tullaghan Well - Tullaghan
St. Patrick's Well - Dromard
St. Patrick's Well - Aughris
The Bog - The Culleens