I like the period after the Aug bank holiday, into Sept. The major holiday season is winding down but all the hostels, B+B etc are still open. The weather can by warm, dry and sunny (as well as cool and wet). The countryside is still green and all the orange roadside flowers are in bloom.
The bit I would like to see in spring is the Burren, the limestone region in Co Claire.
I would suggest the SW western coastline from Cork around all the peninsulars to Connemara, taking some of the Arren Isles. Cork airport is good and there is a train at the end of your tour back to cork. You could get a budget airline ticket from another airport. See Ryanair.
The coast is hillier than inland but much better cycling.