I started tagging all of my data with their appropriate schema.org vocabulary. I'd like my site to display a rich snippet/enhanced listing in Google. Is there ever a case where you would need to include code from more than one markup language (schema, RDFa, Microformats)
Unless you have a reason to use more than one, such a provider of a service requiring a specific microformat to be used, you should generally stick to one format to prevent potential confusion/conflicts between formats. Since search engines seem to be the largest consumer of microdata, and as a result others follow their lead, I would stick to schema.org microformats.
You are confusing vocabularies with syntaxes/formats:
You may use the schema.org vocabulary with RDFa (Lite) and/or Microdata.
Different data consumers (services like search engines etc.; agents like browser add-ons etc.) support different vocabularies and different syntaxes. If you are interested in only one or a few specific consumers, check their documentation and see what they support. Of course it may be the case that consumer A only supports Microdata and consumer B only supports RDFa.
If you like to support as many consumers as possible (i.e. even those consumers that you’ve never heard of or that will come up in the future), you may consider using several syntaxes in the same document.
See also my answer about main differences between Microdata and RDFa.