Sorry for the rough and ready comment on your plans. Any more questions, just get back here or give me a message.
Day 1 & 2 Dublin
Fine but you won't see everything. You never will though
Day 3 Travel to Bushmills
Train to Belfast in just over two hours. Then it is probably an hour from Belfast. It will take you a good 4 hours or more I would say by the time connections are made etc. Look at the timetables though.
Day 5 Giant’s Causeway AM -- travel back to Belfast
Consider getting a car for these few days in Belfast. It will make your touring so much easier.
Day 6 Kilkenny
Belfast to Kilkenny in a day. More than doable, but I personally would not want to do it! Get the train if you really want to do it. It is important to remember that you are going in June. You will have light from around 4am to about 10 or 11pm, so if you really do want to dit alot into a day, you can. But i would advise, normally, take it handy and relax.
Day 7-9 Cashel
1 night is enough in Cashel I think, but if you get a car again, cashel, Kerry, Claire, you would see alot more. Or even just a car for the Clare bit. Personally, I much prefer the Dingle Peninsula.
Day 10–13 Kenmare (Day 11 Ring of Kerry Bus Tour)
Day 14-16 Doolin
Day 17-18 Glendalough
Doolin - Glendalough in a day, good luck! Go to Galway, see it, then get the train to Dublin one morning and get an afternoon bus to Glendalough. Again if you had you own car, Wicklow Mtns are stunning. Don't forget in dublin though, you can do a nice cliff walk around Howth.
Day 19-20 Newgrange
One day is more than enough. A day trip from Dublin is probably better.
Day 21 Dublin