You should report everything that you test in your paper be they significant or non-significant. If you don't, then it gives rise to selective reporting; also known as p-hacking.
What is the goal behind the exercise? Is it just to see which variables are related and how they relate or do you want to predict?
If it is the former then standard practice is to feed in all your variables (i.e. multiple linear regression) and then simply report p-values, effect size and degrees of freedom. This only works well if you have specific hypotheses you are wanting to test and typically you only want a handful to test at a given time. If you are more interested in exploring which variables are related generally to generate hypotheses then I would suggest you look into multivariate methods like principle component analysis (PCA), although this somewhat depends on your data.
If the goal is prediction then one can refine the model (and there a lot of different ways) down to significant predictors so that your model generalises well to new data.
If you provide more information I am happy to help more!