Only one dependent is correlated with independent, now regression and moderation analysis should run for all variable or only one variable which is corelated?
The "correct" analysis, I believe, would depend upon your specific research question and/or hypothesis (as well as the nature of your variables, their quantification, and your sampling or data collection methods). Unfortunately, your query doesn't make that clear, so perhaps you could elaborate a bit. Doing so would likely help you to get more constructive advice from the RGate community.
The "correct" analysis, I believe, would depend upon your specific research question and/or hypothesis (as well as the nature of your variables, their quantification, and your sampling or data collection methods). Unfortunately, your query doesn't make that clear, so perhaps you could elaborate a bit. Doing so would likely help you to get more constructive advice from the RGate community.
Along with David Morse 's advice to tell readers what your research question(s), you should not hide from readers information about your variables and your study. Including your sampling method and sample size.
I agree with the others that a bit more information would be important. I nonetheless give it a try--with regard to the point whether a regression should be done only with correlating variables). The answer is a clear "no" as it is possible (and occurs often) that you'll find a relationship between X and Y in a regression while there is no correlation between both or even a correlation with the opposite sign. Often the latter signals some problems (e.g., a suppressor effect) or the correlation is simply the result of opposing signs of the X-effect and relationships with other predictors.
As you attempt to analyse moderation, it is even more fruitful to proceed as there does not have to be a main effect in order to moderation to take place.
Hence, create a theory based model and proceed. This "theory-basis" should optimally not be restricted to your focal X variable(s) but also include possible confounders of both (or instrumental variables).