My goal in going to Japan is to immerse myself in the language so that I can become much better.

This requires my first trip to Japan to be longer than a month... but I suppose if your purpose for going was just travel and interest then a couple of weeks to a month would be a good amount of time to take it all in.

I voted for one month. I'm traveling to Japan in mid-October and I'm staying for a full month. I figure that I if I have to spend 14+ hours on an airplane getting there, it should probably be for a substantial amount of time.

Also, I'm very keen to learn about Japan, it's history, and it's culture. I would like to move there to teach English or work, and staying there for a month will give me a good idea of whether or not I can handle it.

2-3 weeks should be adequate for a first trip ... if you like it there is always time to come back :)