Epic trip
I think you should go up to the giants causeway, work across to Derry, then down to Sligo, learn to surf then go through the normal tourist spots in Galway/Conemarra/Cliffs of Moher/Ailawee caves and then head down to the beara peninsula via Kenmare
dereen gardens in Lauragh
dursey Island for the only cable car in Ireland
Castletownbere for fresh crab
Ailahees and eyeries for cute little villages
adrigole bay for sea kayaking with seals
Garinish Island for cool spanish gardens
and then move towards Malin head and onto the rest of west Cork and Cork City.
Trip into Fota island
Kinsale for more gorgeous sea food
English market for nice food
In doing this you will do one of the main things to do in Ireland before you die according to national geographic and also dodge a serious amount of hammy tourist spots like tralee.
http://travel.nationalgeographic.com/travel/top-10/things-to-do-in-ireland-and-northern-ireland/
If you spend this much time in these areas you will not regret it. However miniature Leprachaun ornaments may be lesson supply.
There's loads of great music festivals on over the weekends of the summer, all worth checking out particularly body and soul/sea sessions
28 days is more than enough time, flights between Ireland and UK are real cheap if you decide to move on sooner.