General Content Search of Content type Exposed form
This is certainly possible and there can be many approaches to this, but it would require some effort in any case.

Taxonomy terms are not indexed by core search, therefore you would need to rely on a third party module. Such popular modules are Search API and Apache Solr.

My preferred method, in short, is to use the Search API in combination with Views:


Install the modules Search API, Search API DB, Views, Ctools.
Create server and index, add the taxonomy fields in the index.
Create a view for the search page based on the index
Filter by eg. "Search: Fulltext search", exposed
Add any fields to view, such as "Rendered entity" (for teasers etc.)
Sort by any field, such as "Search: Relevance".