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.
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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.
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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).
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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.