Life beyond Earth is, in my opinion, not only possible put extremely likely. As you said, there are trillions of galaxie with hundreds of billion of stars system each. Unfortunately, at the moment, we do not have the capability of directly finding life on exo-planet. Our approach at the moment consists of identifying planet that are in the "Goldilocks Zone," which mean they are the right distance from a star to have liquid water and and other condition that we believe are essential to life, and we have already identified many planets that fall into such a zone. Unfortunately again, we do not have the technology to directly see onto the surface of these planet, so we do not know if there is really life on these planet. So for now, no, we are not equipped to effectively hunt for alien life.
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