Cool, enabling people to do things like this is exactly why we started pushing data out to JSON.
First, a quick outline of the steps you CAN take to develop your additions. Most of these are OPTIONAL, but in my experience, the more people you can get to look at and try out your stuff, the better the end product is.
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Make changes in the JSON files locally. (mandatory)
No way around this one.
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Test changes locally to make sure they work. (optional, but please, please do it)
You might think your changes are trivial, but take every opportunity to verify.*
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Post changes to the forums for feedback. (optional)
This is a big part of why we used JSON, post your changes in [ code ] blocks, and people can copy/paste to their installs and try your stuff out, and give you feedback. Also passing json patches around will give people opportunities to learn how it works, increasing the number of modders, creating a virtuous cycle that cumInAtES In WORLD DOMINATION! MUAHAHAHAA! ahem sorry. It’d be nice if you post it on the forums.
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Make a Github pull request. (optional, but very helpful)
If you don’t make a pull request, SOMEONE will have to do it**, either another forum user, or a developer. There’s no guarantee of either one happening, so doing it yourself doesn’t leave it to chance.
Just making the pull request might be the end of it, or it might not, we’re pretty picky about some things, and might ask you to make some changes. Comments on the pull request will appear on the github site, and depending on your settings you may be emailed when they’re made.
Now to your actual question (how do I github?):
The github help site is VERY helpful.
In particular what you need is set up git and fork a repo, the main repo address to use is https://github.com/CleverRaven/Cataclysm-DDA.git
Once that’s set up, using pull requests explains pull requests.
This might seem like a copout, but there is NO WAY I can explain how to do this stuff better than they did.
I’m aware DBAs need reminders of this less than most, but I want the guide to be general 
If you want to be helpful and import JSON chunks posted on the forums into git and make a PR, look into the git commit flag “–author=” and use it, but feel free to put "git import by: " in the commit message. Maintaining proper attribution isn’t just polite, it’s the LAW!
***No really, the CC-BY-SA license mandates maintaining attribution, so if you strip it, you’re comitting copyright infringement.