Z scores - dos and don'ts

I need to determine whether it would be beneficial to standardize data/ convert to z scores to compare two different sets of data values. Does anyone have a “dos and don’ts” list of when it is best to standardize data/ use z scores and when it is not helpful to standardize data/ use z scores for comparison? For example, I understand that you should not standardize data if the distribution is not normal – is this correct? Any “dos and don’ts” on when to standardize/ convert to z scores would be much appreciated. Thanks in advance.

I would think whenever you can and it is applicable you should use z-scores. By using z-scores, you can use the student's t distribution to run t-test and find out where the standardized score lies i.e. 1 SD away, 2, 3, ....