Hi @Johanndb,
I suspect that you have a FeatureTable[Frequency] as output that you're interested in working with. There are a lot of tools in QIIME 2 for downstream analysis of these - e.g., core-metrics. Most actions that can take a QIIME 2 FeatureTable[Frequency] that was generated for example with q2-dada2 should work with the result of q2-picrust2 (the exception would be actions that require another input that you might not have, like a phylogenetic tree).
If you specifically want to use R for your next steps, I recommend looking into using qiime2R - that's a third party tool that is very popular for working with QIIME 2 data in R.
As for Venn diagrams, we don't have a direct way to generate those in QIIME 2, but this post provides a suggestion.
In general, I would recommend doing a test such as beta-group-significance, which will tell you if your groups are significantly dissimilar from each other in their composition as that will have a clearer interpretation. You could also pair that with a Venn diagram to aid in interpretation of the diagram (Jaccard distance might be a particularly relevant metric there, as it's effectively the sum of features that fall outside the intersection of two samples in a Venn diagram of their features).
Hope this helps!