It's great that you are planning a trip to discover the natural sites of the island!
Public transport is usually suitable to reach most sites, especially when there is a visitor centre. I would suggest using the Transport for Island website
to plan your itinerary and know which places are easily reachable and which aren't. Here is also more information about public transport in Ireland
Since you'll be visiting Co. Galway, I can definitely recommend you the Cliffs of Moher
and the Loop Head Peninsula (Co. Clare) or Connemara National Park and Kylemore Abbey in Connemara. All these places are reachable using public transport from Galway City for instance.
I hope this helps! Please let us know about your itinerary so we can help you find out whether a rental car is necessary or not. 😊