What is the Jewish standpoint on miracles? How important or unimportant is miraculous phenomena to the Jewish believer?
Allow me to rephrase your question in the opposite manner: "What is the Jewish standpoint on nature? How important or unimportant is natural phenomena to the Jewish believer?"
G‑d manages every aspect of creation at every given moment. There are no rules He must follow. There are no forces He must contend with. All is in His hands.
Nonetheless, He chose to create a system called "nature." An arrangement of fixed rules. An order of causes and effects. Why did he create nature? In order to conceal His identity and hide His footprints. He wanted a world in which things would appear as if they run on their own, and thus, force Man to discover G‑d on his own. In fact, the very word for nature in Hebrew, "tevah," also translates as "sunk." Nature is G‑d's way of submerging His presence under a sea of scientific laws and patterns. And Man is a deep-sea diver given the task of finding G‑d's hand which lurks behind the veil of nature.
And so, life is very similar to a game of "Hide and Go Seek." But every now and then, G‑d emerges from His hiding place and breaks through the self-imposed shackles of nature. The sea is split. A scientific rule is broken. Mother Nature is proven wrong. Perhaps, a child is cured from an incurable disease. Or our nation is saved from a seemingly hopeless situation. And it is through these supernatural events that we realize that nature too is merely a creation of G‑d.
For more on this topic, please see The Miracle of the First Night, and The Shushan Files.