Developing Computational Chemistry Software
How do I get into method development?

As was said in the comments, one would join a group that works in this field and learn from them. From up close, there is some derivation from previous results and some trial and error involved. And yes, a PhD-sized investment is typically required.

How do I get into method implementation?

It depends on how deep you wish to go. There are open-source projects to look at and contribute to. There are publicly available exercises on implementing HF and beyond. I would not advise to implement DFT from scratch because the numerical integration grid is both critical and difficult. Same is true for the integrals, use a library for them.

How do I learn to use Program X?

It is helpful to learn one program suite well and learn others later. The reason is that many concepts are similar between the programs. Learning one program comes down to locating good worked examples, reading the manual, attending a workshop if possible. One may for instance attempt to reproduce data from a DFT benchmark, since the supporting material usually has a detailed listing, possibly down to absolute energies.