The so-called rule that forbids starting a sentence with a conjunction is obsolete and nonsense. This is what I found in American Heritage Guide to Contemporary Usage and Styles.
Certainly, starting a sentence—even a paragraph—with and can have a dramatic effect, calling attention to the increased significance of the sentence
However, your example is still not a good writing. You should follow a principle called end focus. According to this principle, new information should appear at the end of a sentence. So I think it's better to use a passive contruction for the second sentence.
We need to have a deep sense of morality. And morality is taught to us by moral science.
EDIT : As StoneyB points out, the moral science could be old information—or even the main topic— in the entirely discourse. But without a larger excerpt, it cannot be confirmed. If so, then an active construction is still fine.
We need to have a deep sense of morality. And it is moral science which teaches us morality