Regression analysis in R - which variables to include?

Hi all,

I am doing causal research between the relation between a dependent variable Y which consists of 4 phases (so 4 dependent variables Y1, Y2, Y3 and Y4) and multiple independent variables X1, X2 and X3.

My first step was checking if there were correlations between the Y and X variables. (Y1 vs X1, X2, X3 - Y2 vs X1,X2,X3 and so on). The results state there is a correlation between Y1 and X2, X3 and between Y2 and X1, X2.

In my second step I would like to do multiple linear regression. My question is: 'Should I include the insignificant variables in my multilinear regression?' and 'how do I chose the variables I should add in my model?'

Kind Regards,