Well, to get you started take a look at this article about psgi
and the three posts it links to (which are essentially discussions of CGI vs PSGI programming. Since you're working with a legacy application it may be difficult porting it into a framework such as Catalyst, Dancer, or Mojo. We recently did some conversions from CGI to FastCGI and it was completely painless. Going from mod_perl ought to be easy a well given what you said (essentially one entry point into the application). It seems like the community is moving away from programming specifically for mod_perl, cgi, or fastcgi and more towards programming inside these frameworks which can work under any of those.
I've been working on a Dancer app lately and have been relatively happy, it's certainly nice not having to run apache on my various development boxes.
In any case, you're likely going to have to really learn how mod_perl does things in order to safely port it to another platform.