I've been reading a lot of these lately in various places, and I keep seeing authors make the same mistakes over and over and over. Mistakes like these can quickly put your story in the massive "lame" category. Similarly, avoiding certain things can certainly put your story in the small "good" category. Having read these things since 1999, I've seen some good, a lot of bad, and many mistake. This is an attempt to enumerate the biggest.
These are some of the more annoying mistakes that an author can make. A world that is internally inconsistant is a world that gets stupid fast. The games and anime themselves don't exactly help matters here, but just because they have an occasional inconsistancy does not mean that it's okay for you to have them all over the place.
Easily the biggest and most widely-abused internal inconsistancy. Guns existing are going to have a very serious effect on your world. If just one character anywhere is packing heat, there had better be a damn good reason why other characters aren't. If guns exist there is ABSOLUTELY NO REASON for Team Rocket (or any similar villains) to not be using them heavily. Team Rocket's guns should be used as early and often as their pokemon. You will quickly realize that it is far more easy to have a world where guns do not exist.
It seems obvious, but don't say one thing in one chapter and then completely forget about it or go against it in some later chapter. I think everyone does this at some point, try your best to avoid it or make it very small and unnoticable. Retcons are lame, readers have long memories.
USE OF ANIME CHARACTERS IN STORIES NOT ABOUT THEM
The best advice I can give here is "don't". Characters from the anime should never be major characters or influences in your story (video game characters are okay). It's undesirable, though acceptable, for them to be mentioned in passing for some purpose. I dislike this, however, because the author's idea of what a character should have accomplished and my idea of what a character should have accomplished are usually different. It makes me seethe with rage when I see Ash Ketchum as the champion of the Pokemon League. Some of your readers will not have the same expectations of a character's future that you will. Respect that.
If you must use an anime character in your story, make sure you get their personality right. (There are obvious and widely-known exceptions to this such as Pokemon MASTER, where the entire point is that everyone has changed. A lot.)
You don't have to make every pokemon center nurse a Nurse Joy. In fact, it's preferable if you don't make anything a Nurse Joy. Nothing screams unoriginality more than the generic pokemon center. Well, okay, that was some major hyperbole. But it is horridly unoriginal.
You do not have to use Team Rocket, though using them as villains is okay. If you do use them, please get their personalities correct. Despite what you see of Jessie and James, Team Rocket should be evil with a capital 'E'. They should carry and use guns if they exist. They should get physical with you. They should attempt to slit your throat. Above all, don't have your two agents and their cat use a motto-poem whenever they make their entrance. No one reads that crap, and smart villains wouldn't loudly announce their presence anyway.
NEW TRAINER FICS
Do something to differentiate yourself from everything else out there. A great example of such differentiation is a fic/novel called Sibling Rivalry*, where the main character finds himself in the center of a world-threatening prophecy that he must overcome. The author makes many of the mistakes I list in this post and the beginning is rough, but his ability to differentiate his story from everything else at the time and his exponential increase in writing skill was more than enough to make his story one of my favorites.
The "avoid using anime characters" rule goes double for new trainer fics; don't be Mary Sue. Video game characters are still okay. So is Team Rocket, but please make them evil.
Many readers dispair upon seeing the character start with some superpowerful pokemon like Mew. I'm actually okay with this, provided that it's worked into the story fairly well. However, the character shouldn't be getting this pokemon from Professor Oak.
* You probably haven't seen it unless you were reading these things around 2000 or so. It still exists, but it is somewhat unfinished and I'm pretty sure that the author died irl. Very sad, it was one of my favorites.
MISCELLANIOUS OTHER MISTAKES
I said it before under new trainer fics but it bears repeating. DO SOMETHING TO MAKE YOUR FIC UNIQUE. You can make every other mistake I list in this post here, but as long as you make your fic unique chances are that people will still think it's good.
Use correct homonyms. Understand the difference between affect and effect.
Pick one style of dialog and stick with it. There are two popular styles of dialog, and they go like this:
"Well," he said, "I suppose that that's enough training for today."
- and -
"Well." he said. "I suppose that that's enough training for today."
I'm pretty sure that both are correct; just be consistent with what you use.
Well, that's all I can think of for now. I hope that I've enlightened you to common mistakes that you might have made or are in the process of making, and convinced you to see the light.