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Whether to standardize the data depends on what method you want to use, how you want to interpret the model and your data structure. For example, when we use some models that are not invariant to scaling, like ridge regression and lasso, normally we would standardize the data to make the model reproducible.
On the other hand, in some special cases, the different scales of variables might be indicators of their real structural effects which are what we want to incorporate. In such cases, we might not standardize them.
And in most cases, the thing that really make a difference is the standardization on predictors, rather than the response. In normal linear regression, we would still have intercept (mean of y) if we don't centralize the response (y). In many literature, people prefer to centralize y as well, then they don't get the intercept.
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