I have a couple of variables with different scales - I don't want to use contingency tables so Is there a way to calculate all the correlations with different methods and then compare them?
You'll have to use different methods to compare correlations. Unfortunately for the cases with qualitative variables, you'll need contingency tables.
Quantitative-quantitative: Easy! Spearmean, pearson, and kendall correlation (scales don't matter).
Quantitative-qualitative: Less easy. You can use a t-test to check the equality of means of the quantitative variable over the modes of the qualitative variable. Or analysis of variance. These are contingency-tables.
Qualitative-qualitative: Less easy. This is a Chi-2 test. Period.
You won't be able to escape contingency tables.