statistical tests for null hypothesis - what if model is non linear?

I am reading the An Introduction to Statistical Learning (Gareth James alii, Springer) as a primer to machine learning.

I am reading the part in linear regressors, and learnt there are different tests for measuring correlations and significance of correlations between predictors (also named variables)- under the assumption that the model may be linear.

What about if the relationship between variables is (or assumed to be) non-linear ?

I also read that anyway many linear models concepts underpins a lot of statistical models.

Can you explain which could be a good practice when one has to model a context that he/she does not know yet?

• Shall we test linearity because it is simpler , and if not found, we'll proceed with other tests ?

• What about though, for testing correlations between variables in non-linear models ?