I take this question to cut broadly - the reason experience is a fertile starting point to think about science and religion is the same reason its a good place to focus our thinking about how we understand ourselves, the world and our attempts to make sense of them both. The unfortunate fact is that in much of modern scientific and philosophical thinking, experience has been forgotten as a question at all. Which brings me to Francisco Varela and phenomenology
Varela was a Chilean biologist of Harvard training who had a distinguished career thinking on a broad range of topics from insect vision to the nature of consciousness. Many people are familiar with his work which has been quite influential (he was one of the founders of the Mind and Life institute) so its a bit embarrassing for me to only encounter him now. Lately I have been reading an article by Varela in "Explaining Consciousness: The Hard Problem". This is a collection of pieces in response to David Chalmers' concept of the Hard Problem in the study of consciousness.
The Hard Problem that Chalmers pointed to was none other than experience - that elusive but inclusive totality that makes consciousness so different than other external phenomena science wishes to address. The text is great because it has so many perspectives in it but for now I want to simply touch on Varela's main focus in his article Neurophenomenology: A Methodological Remedy for the Hard Problem.
In confronting the hard problem Varela reaches back about 100 years in the history of philosophy to the the school of Phenomenology. This was a European movement advanced by writers like Edmund Husserl and Martin Heidegger. The emphasis in Phenomenology is to take take experience and its investigation seriously rather than just dismiss it as unimportant or not really a worthy subject. As Husserl claimed "back to the things themselves" or as Merleu-Ponty wrote
"To return to the things themselves is to return to that world which precedes knowledge, of which knowledge always speaks and in relation to which every scientific stigmatization is an abstract and derivative sign language, as the discipline of geography would be in relation to a river, a forest a prairie we knew beforehand."Taking the phenomenological perspective Varela claims that experience is irreducible. He then goes on to outline how a phenomenology when combined with cognitive science and neuro-science could provide constraints on each other to develop a true scientific study of consciousness.
Varela's perspective is an important one for thinking about consciousness and science in general. Experience is irreducible. It is always were we start and where we must return. The inability to recognize this has led to the objectification of objectification in science. We imagine the detailed nature of an objective world because it is so fruitful for our scientific world-view (and it is) but then we forget that no one ever experiences it. Objectification is a useful tool but it is not a thing in itself. It is a powerful story that helps us make sense of things.
Clearly there is a world out there. No one doubts that unless they have gone loco. The world is there and it kicks back. But what access we get to it, and what role we play in shaping what we see is very much an open question. We always begin from "behind our eyes" so to speak. What makes Varela's perspective in this paper (and his book the Embodied Mind) so valuable is it takes that recognition and then attempts to do something concrete with it in the service of investigation.
Varela takes our experience seriously and that is an attitude we could fruitfully apply many places.