I want to compare differences between two groups(n1=26, n2=18) regarding their performance on different tests but also explore how each group performed on different times of these tests. I am therefore inspecting my data for normality to see if I could use t-tests, one-way anova and repeated measures anova. Shapiro-Wilks testing showed that for some variables group1 responses met the normality criteria but group2 not whereas for some other variables it was the opposite. How should I proceed? Use parametric tests for some variables and non-parametric for those that do not meet criteria or non-parametric for all?
Your sample sizes are quite small, so you probably cannot reasonably rely on statistical tests that appeal to the asymptotic distribution of quantities under large sample sizes. ANOVA tests in parametric models use distributions that occur either from underlying normal data, or by large-sample distributional approximations. These should generally be avoided if you have a small sample size and your underlying data deviates substantially from normal data.
If you have two paired measurements (pre-test performance and post-test performance) for each of the two independent groups, you may try to study the evolution of each individual: $$ e_j = post_j - pre_j $$
Then, you may apply a Mann-Whitney test to compare the two independent groups of individuals with respect to the evolution. (I'm assuming that the evolution will not follow a normal distribution.)
Regarding the interactions, you may start by computing Spearman's correlations between pairs of variables.