I want to run a test in my research to determine if there is a relationship between 2 variables each with 3 levels. Can I use a chi-square test for this or a Kruskal-Wallis or something else altogether? Any advice would be appreciated.
Yes, of course. Chi-square is mainly used to answer these type of relationships.
Yes, Brij said it right. But keep in mind that the chi-square test ignores the ordinal structure of your data. For example, if you permute the labelling, you end up with the same chi-square. Whereas, if you want to take into account ordinality, you could use the linear-by-linear test in the crosstabs output.