I'm starting Japanese as well! Good luck to you.
As to your question, I have a few thoughts.
1. I'd say no order is good. Try to learn them all at the same time. Someone said in another thread that you can learn all the kana in a week. I think that's doable, but it really depends on the person. But there's no reason you can't learn grammar at the same time. And really, the sooner you get started on the grammar, the sooner you'll start learning the language proper, and the sooner you can start applying your knowledge of the kana. As to kanji, someone said in that thread that the best way to learn is to start, at some point, learning the kanji for whatever words you are learning. That is, don't try to remember the kanji and all associated meanings independent of your vocab learning. I agree with that, from experience, so try to learn at least a few each day as you build your vocab.
2. Having studied classical Greek, which obviously uses its own script, I can say that proficiency with a new script only comes when you use it. Rote memorization helps, but your brain needs practice in processing a new method of reading. The only way to do that is to write and read in the script. Now, at the moment I'm learning grammar through romaji, but I write my answers in kana. It doesn't matter if I don't use the appropriate kanji (yet), because all I'm trying to do is acquaint myself with processing the characters. As I learn kanji I can add them in. This is the single best thing you can do at the early stages to learn the kana and build your proficiency in reading Japanese.
@DaisukeJigen. Sorry, no recommendations for audio courses. But another thread mentioned listening to Japanese music as a way of acquainting the brain with processing the sounds of the language. I find it useful. So why not try to find some Japanese music you like, even if you can make out the words yet, and listen to it in addition to trying to spend time doing an audio course?