As an electrical/computer engineer, I already have a relatively thorough understanding of classical electromagnetism.
From what I understand though, classical EM is only an approximation to quantum electrodynamics. I'm very curious about how it all really works though.
So as an ECE engineer, what would be the best way to approach quantum electrodynamics? (assuming taking a course at a community college is not an option)
What is an ECE engineer, an electronic-computer-engineering engineer?
Indeed Classical Electrodynamics is only an approximation to Quantum Electrodynamics.
If you just want to get a taste, I would suggest reading Feynman's QED: The Strange Theory of Light and Matter. It describes the theory quite nicely without too much maths.
If you want to learn full Quantum ElectroDynamics, you're going to first need to learn tree-level (introductory) Quantum Field Theory, which will require quite a lot of time, effort and maths. If this is indeed what you want, then check out the books list question for some ideas of where to start. I don't know exactly how much education you have so far in terms of maths.
It's not easy, but it is a pretty great theory alright!
In order to gauge your level of previous knowledge, you could try Engineering quantum electrodynamics, by Dietrich Marcuse ( https://books.google.com.br/books?id=SGd5AAAAIAAJ&dq=editions:STANFORD36105030208412&hl=pt-BR&redir_esc=yEngineering quantum electrodynamics )
