Yes, awe arises during the extraordinary: when viewing the Grand Canyon, touching the hand of a rock star like Iggy Pop, or experiencing the sacred during meditation or prayer. More frequently, though, people report feeling awe in response to more mundane things: when seeing the leaves of a Gingko tree change from green to yellow, in beholding the night sky when camping near a river, in seeing a stranger give their food to a homeless person, in seeing their child laugh just like their brother.
There is no effort that we can make to still ourselves. True stillness comes naturally from moments of solitude where we allow our minds to settle. Just as water seeks its own level, the mind will gravitate toward the holy. Muddy water will become clear if allowed to stand undisturbed, and so too will the mind become clear if it is allowed to be still.