Varies wildly on the story, and the time of my life, so I apologize in advance for the detailed answer. This is what you get for asking authors to talk about themselves.
- I started off with the strategy of just beginning to write and seeing where it goes. For my troubles, I have nearly a foot of one-page story beginnings and brainstorms. In these, usually I had an idea I liked, or a character I liked, but I really had no idea where I was going, and would inevitably get bored. There were a couple exceptions to this rule, but only a couple, and they usually involved comparatively more long-term planning than the others.
- I wrote a lot of short stories in middle school. For these, I usually knew where the story was going, roughly, but would change and refine it a lot while writing.
- My first serious writing project, a novella for eighth grade creative writing class, I roughly planned out beginning to end. I refined the ending to fit better with its themes, but stuck mostly to the plan.
- My longest project, a 270-something written page behemoth, I wrote in real time: On September 3rd, I would write events that took place roughly around September 3rd. This was fantastic for my productivity, but this story was a hot mess with no plot.
- I frequently come up with detailed brainstorms and ideas I have no plan to seriously write in the near future. This is just for fun, and I don't consider it serious "writing".
- Capture the Flag and Diplomat I first planned out every choice on a google spreadsheet. This was subject to a lot of change; Diplomat originally had 8 paths instead of the final 5, but what I did keep stayed relatively true to the plan.
- My game-like stories (Ruins of Anzar and Secrets of the Crag) I plan first, I already knew the contents of each area and the overall plot before I started coding.
- The non-cyoa story I'm currently working on, I'm brainstorming and writing non-linearly. I'll just add a note or occasionally write a scene anywhere in the timeline, wherever I feel like it. The story arc is planned in a lot of detail, but I haven't begun seriously writing.
In conclusion: My initial instinct was to "just write", but I find the more I plan in advance, the better it goes. Editing is also a big part of my process.
For research, I'll just google something whenever it occurs to me. Nothing really formal. Often I end up writing stories about things I already know, rather than the other way around.
Yeah, I usually write one path all the way through, then write all the "death" endings for it, and then move on to the next one. Sometimes I jump around if I feel like it.
You've probably heard this before, but I don't think there's a "best" approach. Just whatever works best for an individual person. It's going to be different for everyone.
Yeah, strong intro is a BIG deal.
For the mythology content, I'm sort of planning on going the "Avoid research by writing about something I already know" route. I'm already familiar with a lot of greek mythology, so I'll probably work with what I know there, and research to fill in the gaps. (This might be subject to change, since I entered 5 minutes ago).
My most successful stories (i.e. the ones on this site) start with a simple concept, and end with a simple execution, where I cut out as much extra ideas and work as I can. Virtually every story I've published on this site was originally plotted to be at least twice as long, but I pruned it down once I realized how long it was actually going to be. This is important for my stamina as a writer, and the stamina of readers.
My favorite ideas generally come from a concept for an interesting scene, dynamic, or situation, or several of these strung together. Everything else, plot, characters, setting, I build around these central concepts. Don't think I've published anything like this here, except maybe my entry to the Agreena.
My writing style is very 'moment' based. I'll have characters and I'll want them to go through certain things. For example my 'Angel' series of short stories I've published here. One major scene I wanted was to have a fallen angel choke slam someone. The rest of that was making it happen and the reaction.
Most of my research is remembering something then looking to see to how I could apply it. Like I could go oh yeah I want a mythical horse in this, doesn't Celtic mythology have something like that?
I actually don't recommend my method, it just works for me. I wish I could sit down and fully plan but I just can't.