I am currently working on a research that follows the attached framework and I would like to ask for advise regarding what method/s of analysis I can use to evaluate the relationship and/or effect of each variable to another.
There is a difference between correlation and regression, in the correlation you do not choose the role of the variables, in regression you must choose the role. If you use the correlation you can build two regression models, in which at first time the first variable is dependent and in the second is the regressor.
If you are interested in a regression-based approach to moderation/mediation, Prof. Andrew Hayes has some great free resources on this topic (including a macro you can use with SPSS for analysis): http://afhayes.com/introduction-to-mediation-moderation-and-conditional-process-analysis.html.
There is no technical difference between running SEM or regression on the same model -- SEM only makes a difference when there is a "measurement model" involved, which is not the case here.
To do a regression of the type you describe, you would include your moderating variable as a 4th IV and then create interaction terms between that variable and each of your other IV's. I would then start by running three additional regressions that test whether each of the individual interaction terms has a significant effect.
I have a question about models that have one important output but have several stages. Can I use SEM and regression to compare the performance of these two models together? Could you please explain more about the measurement model?
SEM requires a measurement model that links observed variables to theoretical concepts, such as having several variables that are all part of a larger scale. If you are working with your variables directly, then you do not have a measurement model and rely entirely on regression. Technically, regression is just a simpler version of SEM (no unmeasured variables), so you cannot "compare" their results.
If you have several stages in your model, the most common procedure is to run a set of regressions where you sequentially add blocks of variables, according to their location within your larger model. Historically, that was known as hierarchical regression, because of the hierarchical ordering of when the variables entered model. But that label became confused with "hierarchical linear modeling" (which is something else entirely). So, more recently this has been called sequential regression.