Backpacking and camping for 3 weeks in Ireland

Dear Forum,

I will be in Ireland for two months, June1st-August 3rd. I arrive in Dublin for 4 days and then I have three weeks to backpack before arriving in Limerick. I would love to know what I must see. I have all of my backacking and camping gear and am interested in backpacking at Connemara, Killarney, and The Burren national parks. I also would like to hike in the Aran Islands and the Kerry Way. I am having a difficult time finding official campsites. I do not plan to rent a car. I will backpack and take buses. I also want to use my tent as much as possible, but I understand that it is not always possible. Any suggestions and must-see's? Thank you very much.

Good luck with the camping in Ireland!

You might want to check out public transport before you commit to not hiring a car. Irish Rail is the name of the train service while Bus Eireann would be one of the main bus companies. It's not going to be necessarily easy to get to the sights as public transport won't necessarily get to them.

Must sees? Well that depends on what you're interested in.

Dublin - hop on, hop off bus, Guinness warehouse, National museums (free), National Gallery (free), Dublin Castle, Trinity College and Book of Kells, a walking tour, IMMA (free), Phoenix Park, St. Stephen's Green, Botanic Gardens.
Near Dublin - Glendalough if you're into hiking,
Limerick - if you're finishing up there then visit Bunratty Castle and Folk Park (bus every hour and takes 25 mins), Ballyhoura Mountains for hiking and mountain biking, Adare - gorgeous village.

For camping, apart from "official" camgrounds, many countryside hostels have some tent spaces, which is a good deal, cheap accommodation plus access to hostel buildings and easy socializing.
Like in Cashel, Corofin, Letterfrack, Cong, Dingle, Doolin, as far as I remember...
Check IH of Ireland