How can I be a good reviewer?
This is mostly just a whole bunch of comments strung together.

First, I think you should look for advice specific to refereeing math papers, as refereeing in math is quite different than most other fields.  (Though it's perfectly valid to ask for general advice now, at some point you should want to know about refereeing specifically in math.)

Second, here are some places with advice:


Attributes of an ideal referee, Notices of the AMS
Refereeing a paper, from MathOverflow
How do I referee a paper, from TCS SE


One thing I would disagree with in the first article is the statement that "The referee is expected primarily to check the correctness of the paper."  I would say the primary job is to assess the importance of the paper, which includes correctness, though it is not always expected that the referee check every detail.

Anyway, since your question is rather broad, I suggest you read this and other Q&amp;A's here and on MO, and then if you have more specific questions you can ask them separately.