Welcome, you have definitely come to the right place 
You have a couple different options, depending on whether you want to separate before or after taxonomy classification:
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analyze together through taxonomy classification. Use the SILVA 16S + 18S database for classification (we have a pre-trained classifier for full-length SSU with this database, or you can train your own with your primers). Subsequently, you can split your sequences and feature tables into separate tables/sequences for bacteria/archaea and for eukaryota, or whatever taxonomic groups you are interested in, but using qiime taxa filter-table.
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After denoising/otu picking but prior to downstream steps (e.g., taxonomy classification) use qiime quality-control exclude-seqs to split your sequences into two different sets based on alignment to a reference sequence set (e.g., 18S sequences) within a certain percent similarity. I do not know what % similarity would be suitable for differentiating 16S and 18S sequences. If you know of a non-degenerate form of your primers (and primer sequences are included in your reads), or an internal sequence within your amplicon that easily differentiates 16S from 18S, you could use
exclude-seqswith theblastn-shortmethod, which is optimized for searching for very short sequence alignments, e.g, with a very high % similarity to differentiate these groups.
That should do it! I personally might go the former route, if there aren't concerns about the reference taxonomy, as it would be a little more streamlined and explicitly identify sequences that cannot be classified as 16S OR 18S. You could always use that approach to split seqs/tables, and then reclassify those sequences with a different reference database or taxonomy classifier if for some reason you suspect the 16S + 18S classifier would give inadequate results (I have no evidence that it would) or if you have a favorite database that you want to use.
I hope that helps!