I would like to get started using modules and tests, but I don't know how to modularify the code I've already started working on!

Just what do you mean? Have you read these?

Tutorials: Modules: How to Create, Install, and Use: Creating and Distributing Modules: Simple Module Tutorial

Modern Perl: the free book Chapter 9 Managing Real Programs

Testing

Handling Warnings

Files

Modules

Distributions

The UNIVERSAL Package

Code Generation

Overloading

Taint

Use these to start making distributions (easier than dzil), module-starter, scriptdist, Test::XT, File::ShareDir::Install, Devel::Local