I currently have a working redirecting configuration for Exim4 on my server. How do I not only redirect the emails to the places specified, but also save them first locally?

You have to use verb unseen in the routers. Usually the first matched router processes messages and then the routing is complete. However, when matching routers have an "unseen" line, processing continues as if the messages haven't been matched before. The following is the correct router configuration to accomplish this:

vdom_aliases: # redirect message
  driver = redirect
  domains = dsearch;/etc/exim4/virtual
  data = ${expand:${lookup{$local_part}lsearch*@{/etc/exim4/virtual/$domain}}}
  unseen      # keep routing

localdeliver: # do everything else
  driver = accept