### Statistical checks for regression, covariance, etc?

Suppose I have some input vector(s) of scientific data. Assuming I at least know which ones are completely independent, and have some other(s) of which dependence is uncertain; what are some good sage functions to throw at the vector(s) to check for basic statistical parity/significance?

I know this question is very broad, so it's a basic howto rhetorical question. A continuous or discrete first order relationship is a fine intro; though more complex scenarios tend to be interesting.