I started writing anything seriously and consistently when I started writing Buffy the Vampire Slayer (guess which ship) and Angel fanfiction in college. I haven't written fanfic in a year or more due to my own writing and my contract ghostwriting, but it was a big part of my life for about twelve years and where I got my first million "practice" words and, okay, a ton more.
Sometimes I think fanfic gets a bad rap. Some creators in the past, like Anne Rice or George R. R. Martin, have gone so far as to discourage sites like Fanfiction.net from hosting it and to also urge fans not to write it. I think fanfic was hit by this double-edged sword with Fifty Shades of Grey (based on Twilight, duh) and its visibility. It's a poorly written work by any standard but I've read a million better fanfics across a dozen fandoms so it does chafe that this story became the one people in general think of when they think of fanfiction. Also, I think it brought on the idea that fanfic can turn a profit, which we see when One Direction stories on Wattpad get 3-book signing deals and with Kindle Worlds and its sanctioned use of fanfic.
In a post Fifty Shades world, the general public are more interested in it than ever---Hell aware it exists---and seeing the dollar signs with it.
The practice of taking down a fanfiction and repurposing it to be an original is usually either called a P2P (pull to publish), scrubbing, or "filing off the serial numbers." It's basically the process of changing enough from the original intellectual property to avoid lawsuits. For example, Fifty Shades (originally called "Master of the Universe") had all the character names changed to non-Twilight names (Christian from Edward...etc.) and, as an all-human AU (alternate universe) was different to an extent from source material. Anyway, I actually also don't mind effective serial number filing. Some of my favorite Buffy fandom authors have also become favorite original writers but filing and creating stories with their own vampire mythology in place of Whedon's. If you like the work of Sylvain Reynard, RITA nominee Tara Sue Me, or Cassandra Clare's mortal instrument series, to name a few, then you've enjoyed P2P fiction already.
I also think fanfic is one of the best places to learn how to write. Fandoms and, let's be honest, the even smaller shipperdoms within a big fandom (i.e. Like Spuffy shippers within BtVS) are supportive environments where creating individuals are so inspired by a story as to add onto it by art and videos, by writing and meta-commentary...etc. It's a place of offered feedback not just from fans of works or the community but by beta readers serving as editors. Moreover it's a place to experiment. Want to to any tense or any PoV? Then just do an experimental short story. Want to see a pairing you'd never see onscreen like Destiel (most probably) in Supernatural then you can follow that, explore character sexualities often in a way "mainstream" TV or other fiction often won't. You want to try bizarre ideas like crack fic with tentacles or everyone is an animal or a million sci-fi tropes like time travel or de-aging, why not?
And often a community of like minded writers and enthusiasts will be there to encourage you and experiment along with you.
I once worked as a virtual author assistant for a friend from the Maryland Romance Writers who has a ton of success with her work, and, despite my fanfic beginnings, I think even I was surprised to hear she'd started in Moonlight fanfic. I think because she's retiree age but also because even now, fanfic isn't always something ficcers talk about openly. Sometimes, even though I should know better, I feel like I was the only one.
All this said, I also realized when going pro that fanfic is not a one-to-one analogue for original writhing. Pacing is very different between the two formats. In fanfic, you can have a 200,000 word (think almost as long as Order of the Phoenix in Harry Potter) sprawling opus with more quieter reflective moments or conversations than the source material offers. On the other hand, no book for publication will approach that length. Structure is different and world building and original character creation are their own beasts. Skills that are learned and developed after fanfic basics.
That said, if I weren't so busy, I'd still write fanfic as well as my contracts and originals. I can't praise an idea more than I do fanfic because it gets people to express themselves artistically, to have fun in a low pressure way discovering their voice, and it helps build communities and friendships that can last for over a decade (at least in my personal experience).
If you're a fan of a show/movie/book/play/comic/video game/etc. then maybe you want to check out fanfiction too. If you've always wanted to write but starting out with a totally blank canvas is too overwhelming at first, then fandom might be for you. And if you want to find your first supportive writing community and a place to try anything once and as Joss Whedon once said "bring your own subtext," then Wattpad, AO3, or fanfiction.net might be the places for you to start.