Testing affects of non-normal explanatory variables on response variable

#1
Hi all,

I'm having real problems with analysing some data (I am using R), and wondered if anyone would please be able to offer me any advice.

I have 1 continuous response variable (normally distributed), and I would like to look at the individual affect of each of my explanatory variables on this response variable, in turn. I have both continuous and categorical explanatory variables.

To check for normality in these explanatory variables I have used qqnorm norm plots (to check for normality visually), and shapiro.test (to check for normality statistically), and both of these tests showed me that non of my explanatory variables are normally distributed.

Firstly, I need to check for collinearity between my explanatory variables before I can look at the affect of these on the response variable. I had planned to do this by:

-Continuous + continuous = Pearson's test (cor.test)
-Categorical + categorical = chi-squared
-Continuous + categorical = ?? I'm not sure what test is necessary to check for collinearity between these 2 types of variables?

Once I had established that my explanatory variables are not highly correlated, I was going to use Pearson's test and the t-test to look at the affect of my continuous explanatory variables on my continuous response variable, and use the one-way anova test to look at the affect of my categorical explanatory variables (or the aov.test if the levels in my categorical variables were balanced).

However, now I've come in to a real problem, as my explanatory variables are not normally distributed. From what I understand, all of these tests that I've mentioned above, rely on normally distributed data for both variables?

I've tried both logging and square rooting my explanatory variables to see if these transformations would normalise the data, but they do not.

If anyone could offer me any help at all about
a) how to check for multicollinearity in my explanatory variables
b) carrying out analysis of affect of explanatory variables on my response variable
...it would be much appreciated. Thank you very much in advance :)