How wonderful to hear you are planning on heading our way. You are wise to try and stick inside the one area, 5 days is not really long enough to be travelling all over the island.
My thinking would be to make a few stops on the way to Galway, it would break up the journey and give you a chance to see some lesser-known spots, than the usual coast run. This trip idea is for 4 days
but you could pick a couple of destinations from it to get you started.
Once you have reached Galway there are 2 options, explore the surrounding coastline. This is an excellent guide for the bay coast around Galway
Alternatively, you may want to explore Galway it's self. You can read what Galway has to offer here
At this point, I have already given you more than 5 days worth of exploring but you did mention wanting to do the Cliffs of Moher and the Aran Islands. If you do plan to squeeze them in, perhaps a tour, for those locations, would make things a little easier. You can find a list of available tours here.
As a final point, if you are looking for accommodation, I would always recommend going with Tourism Ireland approved accommodation.You can see a list of available accommodation here
Hopefully, between all this you should have a good starting point for planning your Itinerary. I am sure it will have raised as many questions as it answered, so feel free to ask away if you need anything further.