Hi
I agree with Richard's recommendations and these are some of my favourites.
Kerry - it's all great but I would concentrate on the lakes and mountains around Killarney. 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. Also great golf courses in Killarney, Tralee and Ballybunion has (they say) the best links course in the world. If you like the Kerry landscapes and want more then continue along the Cork coast.
Clare - use the car ferry from Tarbert to Killimer to cross from Kerry to Clare. The Cliffs of Moher and the Burren are both worth a short visit, and the unique flora of the Burren should be in bloom in May.
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.
May is a good time to visit Ireland and the west coast is in my opinion the best part of it. Staying 4 to 5 weeks will give you the opportunity to explore most of the island but keep in mind that you will be travelling mainly on slow country roads to get to the best landscapes. At that time of year you will be able to book accommodation as you travel, get a good directory from the Irish tourist board, I found the B&Bs excellent. Do take good waterproof gear for yourself and your camera as the west coast does have periods of heavy rain even in summer.
Have a good trip.
Kathy