The pantheistic mainstream asari religion is siari, which translates roughly as "All is one." The faithful agree on certain core truths: the universe is a consciousness, every life within it is an aspect of the greater whole, and death is a merging of one's spiritual energy back into the greater universal consciousness. Siarists don't specifically believe in reincarnation; they believe that spiritual energy returned to the universal consciousness upon death will eventually be used to fill new mortal vessels.
Siari became popular after the asari left their homeworld and discovered their ability to "meld" with nearly any form of life. This ability is seen as proof that all life is fundamentally similar. Siari priestesses see their role as promoting unity between the disparate shards of the universe's awareness.
Before the rise of siari pantheism, asari religions were as diverse as their political opinions. The strongest survivor of those days is the monotheistic religion worshiping the goddess Athame. Like the asari, the goddess cycles through the triple aspects of maiden, matron, and matriarch.