From personal experience, the best way to develop your fanfiction writing is to focus on a fandom (i.e. Tamora Pierce) that you have a substantial interest in and knowledge of. Your interest in the TP world will be the general source of your inspiration to create a written piece, and ultimately the whip at your back that pushes you to continue writing.
Knowing your focus
You do not need to have a deep knowledge of all the series, all the characters and all the pairings/shippings that are created by TP or by other fanfic writers. It's a bonus if you actually do, but once again it is not essential in building a workable story. What's recommended is a sound understanding of the focus of your story; whether this is an event, a character or characters, a pairing, or a mix of these. It is this understanding that makes plot construction and writing chapters easier to accomplish.
Writing isn't a matter of being the omniscient viewpoint; it's about going into the mind of the characters and using your knowledge/opinion of them to generate the story line.
Characterisation
Good characterisation is the ultimate decider of the solidarity and success of your story (regardless if you're just a beginner writer or an esteemed veteran). There is no story without characters — it is characterisation that drives the plot. This is important when considering if your fanfic will be based upon the canon or not. For example, the character's personality must correspond with the setting of your story. This could mean that you may have to quirk some of the more notable features of the character's otherwise normal (as created by TP) persona.
The same goes when introducing and fleshing out your own new characters. Contemplate how your own created character will get along with one of TP's characters; if there's a clash you may have to change things here and there. The bottom line is…don't be scared to go into the minds of the characters you're writing. Usually, this gives your writing more depth, and gives the characters more than just a one-dimensional feel.
Writing — spelling, grammar, punctuation
We all have to learn to write properly sometime. This is usually in elementary school, and we continue to pick up new things as we go through high school and beyond. However, when writing, it is best that you spell correctly, use accurate grammar and utilise proper punctuations. This determines the flowing quality of your writing. At the end of the day, you are writing for you, and it only makes sense that if you wish to improve your writing, you must write with your best each time.
Using the aid of BETA readers, and/or requesting the advice or proof-reading of someone else is a great way to learn to develop your talent. However, these people can only do so much for you. With or without their help, you must continue to practise your writing and read your own research on how to improve.
Your own personal style
Every writer is different; the approach, the point of view, the interest and the limitations. You will find that as time passes, and your writing accumulates, you will fall into a pattern of writing that suits your abilities, and allows you a greater freedom to experiment. However, experimenting is better to be attempted once you feel that your original focus has been amply polished and approached.
Personally, I found that concentrating on My Focus has improved my writing over the past few years. Flexible and deep knowledge and understanding of the characters, the shipping, and the TP series I focus on has evolved my characters, refined my stories' dialogues, and has improved my grammar and punctuation and widened my vocabulary.
One of my favourite authors on the internet advised to “write, write, and write”. That is the best way to improve your writing, no matter how long it will take (usually years). You will run out of ideas, and you will run out of patience. It's all part of the job, just like any other. There may/will be times when you believe that it's time to quit, but believe me; writers that persevere and continue (even if a story takes forever to be completed) are the ones who get better later on.
Fanfiction means Imagination and Freedom
It doesn't matter if your fanfic is a one-shot or a multi-chaptered story. It doesn't matter if your chapters are 200 words long, or 5000 words long. These all represent a creative part of yourself that you wish to showcase. Every word contributes to the goal you had in mind when you published your work on FF.net or any other TP fanfiction site. Suggestions you may reflect on:
Writing is something we all struggle to get right. Be open, honest and realistic when it comes to your writing — this usually leads to experimenting and improving. Along the way, your writing will get better if you understand what/how/why you are writing. The magic rules are to be interested, choose a focus, understand the focus, make your characters come alive, check your spelling/grammar/punctuation, always persevere…and just WRITE. Nora Roberts once said, ‘I can edit a bad page, but I can't edit a blank page.’