The problem might be there aren't that many people here with experience of immigrating to Japan.
Visiting Japan
Yes you'll need a passport. I presume you're an American citizen so you don't need a visa in advance you'll get one for 90 days on landing. You fill in a landing card and customs declaration, get photographed and fingerprinted maybe speak to immigration officer and usually speak to customs officer. (everyone stops at customs, unlike Europe where you keep going (green channel, blue channel) unless stopped or if you have goods to declare in the red channel)
Getting a passport? In the US the State department handles this.
http://travel.state.gov/passport/passport_1738.html
Travel insurance is a good idea as well.
Google for flights to Japan, check airlines sites. I imagine it'll cost over $1500 depending on where you're starting from. There are 3 main airports Kansai (Osaka), Centair (Nagoya) and Narita (Tokyo). I've never flown through Tokyo myself. Osaka and Nagoya are very pleasant with good transport links. If you are travelling about anyhow it's worth considering using these airports and taking the shinkansen to Tokyo.
Hotels
These sites should give you an idea of what's available depending on your budget.
http://travel.rakuten.co.jp/en/
http://www.hostelworld.com/countries/japanhostels.html
there's a large variety of accommodation available in most places in Japan.
Cost.
hard to say. depends what you do, depends where you stay.
I'd allow at least 5,000yen per day after accommodation costs.
Internal travel.
If you're going to go between Kyoto and Tokyo (or beyond) the Japan rail pass is good value. If you're only going for a week, you might find it better to stay in one region though. 2 weeks or more I'd travel about a bit.
http://www.seat61.com/Japan.htm
timetable search including fares
http://www.hyperdia.com/
other sites
http://www.jnto.go.jp/eng/index.html
http://www.japan-guide.com/e/e623.html
Emmigration
The grass isn't always greener.
Japan is a conservative country by and large. I think there is a fair amount of scope for culture shock and feeling isolated because of an unfamiliar language and culture and probably being functionally illiterate too. Especially if you have no experience of foreign travel before. It'd be a good idea to have some friends or contacts in Japan to help smooth things out.
You will need a visa to work in Japan. To get one you'll need an employer to sponsor you. I'm pretty sure you have to apply for the visa from outside Japan.
Check with the Embassy.
http://www.us.emb-japan.go.jp/english/html/
If you have an undergraduate degree, it seems the biggest potential job market for English speakers is teaching English.
If you're under 30 (I think) the JET program is worth a look as a way to live in Japan for a couple of years. Teaching isn't an easy option though. like most things you need skill and aptitude and an interest in teaching.
http://www.us.emb-japan.go.jp/jicc/japanjobs.htm
good luck.