My question is what language(s) should I learn that will help me build web apps, database apps, and desktop apps besides Perl, in which I can make an easy(ish) transition from Perl?

Many/most languages can do those things. In terms of an easy transition from Perl, I suggest Python and Ruby.

I'm sorry if this is a bit vague, but I just need a recommendation.

Why? For example, is there something specific that you cannot get Perl to do? Or is it to improve your job prospects? Or is it something political in your workplace? Or something else?