In a dataset, there are 2 binary variables and 3 continuous variables but on different scales. I am wondering, if I want standardized coefficients for these variables, do I need to standardize all three 3 continuous variables or just standardizing part of 3 variables is fine too to run regression models? In addition, do I also need to standardize Y when do modeling?
See this answer about some of the effects and changes in interpretation when standardizing your variables. If you want to compare the beta coefficient of your continuous variables with each other you want to standardize them all. Whether you want to standardize at all depends on your goal, see this post. Some issues to consider are whether you care about comparing effects of different predictors within your sample, outside your sample, or perhaps you use standardization as a tool to deal with multicollinearity. To name a few.