How do I start getting noticed as a writer?

[RedStar]

It depends who you want to be noticed by. If you want to be noticed by people who might actually publish your work, then you'll have to approach them - they won't come looking for you. They already have enough people approaching them proactively with publishable work - they don't need to go looking for it. So, you'd have to research the types of publication that publish the type of writing you do (you don't say in your question whether you write fiction or whether you're thinking more of non-fiction articles) and approach them with your work.

If you mean that you don't want to start submitting your work for publication but just want to get some feedback, then your best bet would probably be to take a creative writing class or join an online critique site where writers critique each other's work.

There are actually lots and lots of sites where you can post fiction, or you could start your own blog, or you could post fiction on a deviantART account. However, that isn't usually the best way of getting useful feedback, as a lot of the people who comment will know absolutely nothing about writing and won't be at all well-placed to judge whether your writing is publishable.

I'm not quite sure what you mean by 'things that could possibly turn into quotes'. All writing is quotable, and being quoted should hardly be your main aim in writing.