Basically science is about trying to discern the facts of the world and understand them. For that you need "judgement" - and judgement is mostly what many of the Jewish religious texts - and the Jewish religious institutes were about. So the two are tied in tightly.
The book of Koheleth in the bible attributed to king Solomon - from 900bc (while some say it is from 200bc) discusses knowledge and understanding of the world - being pessimistic about the ability to ever truly grasp it fully, but stating the importance of knowledge - "the knowledgeable has eyes in his head, while the stupefied one walks in the darkness".
About the year 200 The Babylonian Talmud discusses medicine (which is a type of science) trying to decide if eating a dog that bit you is a valid treatment against rabies (not rabbis) or is it "Darkei Ha-Emori" the way of Emorites - in other words a superstition only, with ties to idol worshiping. The Talmud is full of both myths and superstitions on the one hand, and "scientific" discussions of facts like this, on the other.
About year 900 Rav Sa'adia Gaon was the first to write a full fledged treatise as to the Jewish religion and "science" of the day - mostly Greek philosophy, anatomy and medicine, and Babylonian knowledge of trigonometry, algebra, theory of music, with the most important of all: astronomy.
Around 1100, he was followed by the Rambam - Moses Maimonides who, aside from being a doctor, wrote many books in Arabic about science topics. His famous (and only) Hebrew book begins with "The book of Science" and begins by describing the four elements (Fire, Water, Earth and Air).
Haim Ibn Zakuto was a Geographer and Cartographer . He wrote a map for Colone (Christofer Columbus) and his team, funded by Jewish merchants. According to the map, based on a "printing" mistake (actually a copying mistake) which said that the world circumference was only 15,000 miles, it would be quick and easy to reach India by going west rather than east. Cheaper too because you don't have to pay all the stations and pirates on the way, or meet the enemy (Spain was the enemy, when the idea was proposed to the Portuguese king, Portugal was the enemy when the idea was proposed to the Spanish king, who later accepted it). India had spices which could be sold in Europe for a lot of money, and it was worth it.
They set sail accordingly, on the day of expulsion of Jews from Spain, Tisha Beav 1492. Many on the ship were in fact Jews (forced into Christianity) or their descendants and there are various indications that Colone himself was the same. They almost turned back because of the extra 9000 miles not accounted for, but 3 days off schedule and after a mutiny, when the sailors were afraid they would fall off the end of the world, they finally saw India - actually Cuba.
Baruch Spinoza in Amsterdam was deeply affected by his scientific studies, and wrote a whole new idea of living in a Godless world. Taking on this way of thought became prevalent in Germany under Moses Mendelsohn and the Haskala movement, and later on by many secular Jews around the world. Jonathan Miller in a BBC documentary said: The Jewish religion is about eliminating false gods. We just took the Jewish idea one god further.
Many contemporary Jewish thinkers disagree. In Israel, 90 years ago, at the beginning of world war I, Rabbi A.I. Hacohen Kook, began writing his book on science and especially evolution theory (as he understood it then, as a teleological theory - that is: a theory that claims the world is moving and developing towards a goal) as an indicator of a foundation for Jewish ethics. He was following two major thought trends in the Jewish tradition. The first - around year 1700, that of Rabbi Dov Bear of Mezritch, the founder of Hassidism as a movement (he was the student and appointed successor of the Baal Shem Tov), who spoke of studying nature and science as the "Truth" which one must seek, since "God's stamp is Truth". And the second which Rabbi Kook was following was that of the German "Hochmat Israel" (Jewish Studies) - a trend under the Haskala movement, using philology (comparative study of texts), archaeology, and scientific history studies, to understand the original meaning of texts and content.
At the same time there was a strong tendency by religious Jews to move away from science, sometimes forced on them since the Napoleonic era, when Napoleon captured most of eastern Europe and created the "emancipation" laws - where Jews had equal rights. And so we find today many Orthodox and especially ultra-orthodox rabbis with an "anti-scientific" way of thought, or at most with an attitude of reconciliation between the two, with the Jewish traditions especially about the creation of the world, usually prevailing while the scientific view being skeptically scuffed at. - in a similar was as Christian creationists.
Interestingly, there is also a strong connection the other way - from the Jewish religion to science.
Religious and secular Jewish scientists have made a major impact on scientific theory and discovery, many times with a strong relation to the Jewish religion - for instance Jewish medical researchers were dominant in the discovery of Insulin, a cure (- so to speak) for diabetes, the development of streptomycin and other antibiotics, the understanding of the Mitochondrea and much more - due to their seeking for a unified understanding of the world, and viewing medicine as an important deed that humans must participate in. The Jewish notion of cleanliness and especially the ritual washing of hands before meals and after using the toilet, and the ritual baths were instrumental in the adaptation and use of hygiene and sterilization in medicine (as well as saving many Jews lives during the black plague). There were many non Jewish "scientists" who opposed this at first. Read about the ordeal of Zemelweis.
Albert Einstein was only one of many Jewish physicists and mathematicians who made a strong impression on science 70 years ago just after world war I. His whole life he was caught up with the notion of the world being actually some kind of unity, one single rule, one single particle (the proton or quark). This was strongly related to his studies as a young child about the Jewish religion. Many Jewish great mathematicians and developers of statistics and game theory have a deep association with math in the Jewish tradition, such as Gimatria and the "Remez" field of study.
The "big bang" theory caught on strongly and quickly, because of Jewish scientists who were happy to re-conciliate Biblical texts with scientific fact. - Well, how long was the first day, when there were no planets not even heaven and earth, just light and darkness?
Archaeology, scientific text studies and much of linguistics (even anti-Israeli Noam Chomsky's father was into it due to his Hebrew studies with his father Zeev) have a strong following by Jewish scientists, who have a feeling of membership in the field - "this is ours". You can trace the seek for Adam and Eve in paleo-anthropology theories, where early huminids passed through Israel from Africa before dispersing to the rest of the world. Or "The four mothers" study, which of course refers to the Jewish four mothers (Sarah, Rebecca, Rachel and Leah)
The same is true in political sciences where Tikkun Olam (correction or fixing of the world) has had a strong Jewish influence. Professor Leo Strauss' ideas in philosophy are strongly connected to Jewish texts and the Jewish religion in his Jerusalem and Athens theories. Even Chomsky's extreme ideas on ethics can be shown to be closely related to the Jewish religion (which he opposes).
At the same time, this also works the other way - the big political movements had a strong influence on the Jewish religion. Communism was invented by the Jews Carl Marx and Engels, writing in what they figured was a scientific way and the ideas of Capitalism in Jewish "Ein Rand's" books, were spread to Jewish Milton Friedman who is a known Professor of Economic sciences.
Feminism, and feminist studies were strongly advanced in the US by prominent Jewish women, many of them discussing their ties to the Jewish religion and its ideas.
A much more boring discussion would be about science and technology in Halacha. Jewish scientists who are also rabbis are prominent in the decisions about the moment of death, science has been used to determine if there are "worms" in lettuce, and to create cherry tomatoes (invented and developed by Orthodox Jew) so that they can grow disconnected from the ground, and thus be permitted to be grown on the Shemitta - seventh land-rest year.
Advanced timing clocks were developed for turning electricity on and off on the Sabbath day, and other quirks have developed, so that the Jewish tradition can continue, while living in the modern world. For example, today there are medical procedures added to food rituals, and even to circumcision.
Science today strongly affects religion in many subtle ways. The large reform Jewish movement is lead by scientists, and the same was true for the first Jewish Conservative Movement leadership - who were all "Rabbi Professor" or "Rabbi Dr."
I hope I answered your question.