I'm trying to assess correlation between a few different variables - some are numeric, some are presence/absence (labelled 1,0). I've done some googling but can't seem to figure it out being a stats novice - are there any tests I can use to check correlations between non-normal and binary data? Really appreciate any help
If by correlation you mean a measure of goodness-of-fit of a specific class of curves (like Pearson correlation for linearly related variables), you can use Pearson correlation for non-normal data. As you can read here, the normality assumption for Pearson correlation is important for the calculation of p-value and confidence intervals. For the $r$ value specifically, normality is not required. However, you can use Spearman and Kendall correlation, which the normality assumption is not required for the calculation of p-value and confidence intervals, and also extend the class of functions to fit your data, that is, monotonic functions (which includes linear). It's important to say that Spearman and Kendall correlation work on continuous, discrete, and ordinal data, which means that your "Yes/No" or "Presence/Absence" must have an ordering. From Presence to Absence, for example.
Binary data with these rank-based correlations (Spearman and Kendall) can be tricky due to ties, but in some implementations such as in the $R$ language and its common packages, this is taken into consideration.
However, if you mean correlation in the general sense, a statistical dependence measure, how related are a set of variables, then you can use Mutual Information. MI will provide you a degree of relationship (statistical dependence) between two variables regardless of the form in which such relationship occurs. There are many $R$ packages and other software that can calculate it for you, which is pretty straightforward for discrete variables and a bit trickier for continuous ones.