Is it appropriate to run correlations first and then a regression? In my study I have several predictor variables as well as dependent variables/outcomes. I don't have much theoretical pressure to choose any variable over another, making decisions about regressions frustrating. So I ran the correlations between all variables, IVs and DVs to see which had relationships. Is this frowned upon? Can I use this information to inform where to look further?
Are correlations susceptible to familywise error?
 A: In my opinion, it is OK to inspect correlations first. In fact such exploratory data analysis is important, for one thing so that you know about any possible problems in advance with multicollinearity.
The best way to choose covariates in the first instance is by recourse to a priori understanding of the causal relations between the covariates and the outcome(s). A good way to do this is by drawing a causal path diagram or directed acyclic graph. This has the advantage of allowing the identification of potential confounding variables that should also be controlled for, but also the identification of a minimally sufficient set of covariates, to avoid over-adjustment (which can result in the reversal paradox). An excellent description of these pitfalls can be found here.
If you really have no a priori knowledge to help choose candidate covariates, then you are running the risk of choosing covariates on the basis of spurious correlations. In this case you can employ stepwise procedures to choose covariates, but this can result in inflated Type 1 errors (ie familywise error), and you also need to be very careful about multicollinearity.
