One quick and easy way to look an nonlinearities would be to generate a correlation plot matrix (x vs. y plots for all pairs of variables). In Matlab, just use corrplot(). If there are any obvious nonlinearities, look into variable transformation to linearize. This

NIST Handbook is a good, easy-to-read resource.

If you have only one response variable and many more observations than variables, then you could use multiple linear regression to identify important factors. Start off with a full model, incorporating all variables (and interaction effects if you like). Look at the confidence intervals for the coefficients - if the interval includes zero (at whatever percentage limit you are comfortable with; 95% is typical), then you can conclude they are very likely insignificant and exclude them in the next iteration. R-squared can be made arbitrarily high just by adding more terms to your model, so it's not a good metric to look at on its own.

If your variables are highly correlated, you should look into PLS to get more robust results.