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.