"Bad news" means news about "bad" things like accidents, death, illness etc. People tend to tell this type of news quickly. But "good news" (passing an exam, winning some money, getting a job etc) travels more slowly.
We should be careful in a dangerous situation. It is better to stay safe than to have an accident (and be sorry about it). The full "grammatical" sentence would be: "It is better to be safe than to be sorry."
It is better to take a bad step in walking than to say the wrong thing in talking. We can do a lot of damage when we say something by mistake. The elliptical nature of this saying may make it difficult to understand. The full version might be something like: "It is better that your foot should slip than that your tongue should slip."
This proverb drops the verb "to be" (It is better to be untaught than ill taught). We understand: "It is better not to be taught at all than to be taught badly." It's better not to learn something than to learn it badly.