Welcome to Mi Yodeya and thank you for persisting with your excellent question.
Especially in consideration of recent current events, as you have already commented about concerning the sad and terrible synagogue shooting in Pittsburgh, it is highly admirable and praiseworthy that you come forward with your question now.
It demonstrates a desire to show unity with your people, even though philosophically you may not agree with them about everything. And also that we all (not just your own people) must get along with each other in harmony. That appears to be the underlying foundation of the one practice that you seek.
As a self-professed atheist, the content of your question, that you seek suggestions of what would be doable/achievable by you personally, also indicates your character. That you try to be realistic about your nature and apply a rigorous, intellectually honest, measure to your own life.
With that in mind, if you are seeking one, single practice, what is called in traditional Jewish terminology a single mitzvah/commandment that you could keep, it seems appropriate and logical to seek something that is comprehensive and all inclusive in its scope.
It should also be something that you can relate to intellectually, emotionally and spiritually. Ideally, it should be something which you can study about in depth over time and reflect upon again and again, which will lead you to positive growth as a human being for the rest of your life. This is the concept of good in Judaism, that which leads to an increase, profusion and diversification of life.
With all this in mind, a famous story found in the Talmud, tractate Shabbat 31a, concerning two of traditional Judaism's most famous Sages, Hillel the Elder and Shammai comes to mind. An individual with essentially no knowledge of Jewish practice and teaching came before Hillel seeking his help and advice to find this same type of thing. He was seeking the whole of Torah, a single mitzvah so to speak. And Hillel replied to him:
That which is hateful to you do not do to another; that is the entire Torah, and the rest is its interpretation. Go study.
As is explained by one of our later Sages, Rabbi Shneur Zalman of Liady, in his book, Likkutei Amarim Tanya, chapter 32, This single mitzvah, called Ahavat Re'im (Love of neighbors) which is based upon the expression from our Torah (VaYikra 19:18), Ve'Ahavta le'Rei'acha Kamocha which means:
Love your fellow as yourself.
demonstrates that, in reality, we are all brothers (as in brotherhood, not sexual characteristic) with a single, common Father. (It is worth noting that in Hebrew, father can also mean source.)
The beginnings of the details of what this particular mitzvah practice entails are discussed by Rabbi Schneur Zalman in that chapter. It is worthwhile to read it carefully, reflect upon it and try to incorporate its performance into your life.
Further intellectual underpinnings about this mitzvah and its importance are also found at the following link discussing Loving a Fellow Jew and Loving G-d.
May you be blessed with success in pursuit of your goal here. And I am sure, like myself, we are all looking forward to your continued positive contributions to our community.