My research process is haphazard at best, but there are a few common threads that I think could helpful if anyone out there is attempting some amateur research of their own.
Every idea begins with a single thread, and my goal in research is to pull on that thread until I get to something interesting. So sometimes that's a single article, or a website that someone mentioned, or a little known factoid I found on Twitter or Wikipedia. When I have an idea like this, I create a new draft post in WordPress and jot down a paragraph or two about it. That bit is important. Just giving it a title isn't enough, I'll forget it later if I do that. I take a bit of time to write down a good description of exactly what the idea is.
I go in with almost no preconceived notion of what the story is going to be. I just start searching through the web. Every link I collect, I index on Pinboard, along with some notes about the link. Take notes copiously. You never know what's going to be important later, so it's better to throw a bunch of quotes into the same place so you can view the whole
Slowly, but surely, an idea comes into focus. I let that guide the ultimate story. And then it's just a matter of writing a draft, destroying that draft and rewriting it, doing that maybe one or two more times, then publishing what I've got.