Ken,Just depends on what you like. You can always find something along the road anywhere. Go any direction from Dublin and you won't be disappointed. If you do spend time in Dublin, do the Red Bus hop on Hop off. Gives a nice overview. The street buskers on Grafton St are always entertaining.
I spent most of my time in Mayo. Crough Patrick is an interesting hike to do. Some nice bike rental/trails in the area also. The Aren Islands are interesting, but renting a bike there is the thing to do....hire a cab instead. Too many hills and too much wind for poorly maintained bikes.
Not in Mayo, but Cliff of Moher are cool to see, but you will see the other side at the Aran Islands, TO ME it appears they split off from there.
Ring of Kerry is another pretty drive.
Download google maps offline for you phone for navigation. Can you drive a manual transmission?
I have no suggestions for housing as I just stay with family.