Direct answer
The best way to save web fiction for an e-reader is to export a clean EPUB with a sensible title, author name, chapter order, and optional cover image. That gives the reading app or device the metadata it needs for a stable library entry.
Start an e-reader EPUB
Paste the first readable chapter URL to continue in the binder, where the EPUB is assembled with stable title, author, and chapter order.
Why EPUB is the right default
EPUB is easier to transfer and catalog than a pile of open browser tabs or copied documents. It keeps the chapter order intact and works across most reading apps and e-readers.
Next step after export
Once the file is ready, follow the EPUB to e-reader guide or the Send EPUB to Kindle guide.