Hi
June is a great time to visit Ireland and the west coast is in my opinion the best part of it. For Ireland you are planning on covering a lot of ground in 15 days as you will be travelling mainly on slow country roads to get to the best landscapes. Do take good waterproof gear for yourself and your camera as the west coast does have periods of heavy rain even in summer. My favourite locations are as follows:-
Kerry - it's all great but with limited time I would concentrate on the lakes and mountains around Killarney, the gardens at Muckross House should be in full bloom. Then to the coast, the whole South Kerry coast is wonderfull but do take in Kenmare, Waterville and the Dingle Peninsula. If you have time and the weather is OK take a boat trip out to the Blasket isles. Also take the road across the Dingle Peninsula from Dingle to Kilcummin over the Connar Pass, spectacular views especially at sunset. Great seafood and Irish music in Dingle.
Clare - use the car ferry from Tarbert to Killimer to cross from Kerry to Clare. As others have said the Cliffs of Moher and the Burren are both worth visiting, the unique flora of the Burren should be in bloom.
Galway - for a taste of Irish city life Galway City has a lot to offer. Connemare offers great scenery both coastal and inland, just explore the little roads down to the little fishing villages. Cliften is a good location for accomodation. A visit to Connemara National Park is worthwhile if you are prepared to take the walk to the top of the hill, the views are stunning. Inland the lakes Lough Mask amd Lough Corrib are very pretty.
Mayo - Westport offers a good choice of accomodation, resturants and music. Clew Bay is said to have an island for every day in the year. I have been there twice and both times the Bay was shrouded in mist. If you want cliff scenery and long golden beaches then Achill Island (connected to the mainland by bridge) has some of the best. There are some great walks one taking in a deserted village offering some great photo opportunities.