The title pretty well summarizes the question. さて vs. そして: When should you use one or the other? How are they different? To me, they come across as being used very similarly, at least when at the beginning of a sentence, but even then, there are probably different little nuances.
さて is a "decorative" word and is used like "Now,.." or "So, ..". It doesn't really matter even if you omit it.
さて、始めましょうか。: So, let's get started.
そして is a coordinating conjunction and can be translated into "Then," or "After that".
空が暗くなって、そして 雨がふりだした。: The sky got dark, then it started raining.
Also, it could simply mean "and". For example,
私は新聞と傘、そして チョコレートを買った。: I got newspaper, an umbrella, and a chocolate.