My interest in writing arose from fanfic. Then I read a blog where an author ranted on it, basically saying, "If you think you're so good, why don't you try original fiction?"
So I did.
And I am soooo grateful for the challenge.
Fanfiction helps you learn to write by taking at least one element out of the writing process--character, perhaps setting. For me, it was like training wheels on a bicycle.
The cons are that fanfiction is uncopyrightable, and you may be in violation of someone else's copyright if you post it somewhere. If you hope to publish one day, I'd choose to either not post fanfic online (and then, what's the point?) or write under a carefully guarded alias.
And frankly, I've found fanfic peer critiques not very helpful at all--mostly praise, in my case. A good writing group is better.