To work full-time in Japan, you first need to have at least a Bachelor's degree in any field (or prove you have at least 10 years of experience in some industry), then find a company or school that agrees to hire you. You can apply for a visa in or out of Japan, but to change a landing permit into a work visa, with very few exceptions, you need to get it at a Japanese embassy or consulate outside of Japan. Many people come to Japan on a 90 day landing permit (in many countries now, the "tourist visa" is outmoded--also the Japanese Immigration may ask to see your air ticket to show that you don't plan to illegally overstay your landing permit), find a company/school that will sponsor them for a visa, wait around 3 weeks to get a "Certificate of Eligibility", then leave Japan to get the visa, which is typically valid for 1-3 years. (Renewals can be done in Japan). Most people fly to Korea for this. You can also get a Working Holiday Visa, if you're Canadian, British, Australian, New Zealander, or German (no, Americans are not elgible). One can also enroll in a language, martial arts, or other accredited school and get a student visa which allows you to work up to 20 hrs/week legally (there is a separate form from Immigration you'll need to fill out). To get a student visa however, many times the Immigration Dept. will demand that you have at least one million yen in a Japanese bank account, and to see your college diploma (the real one or certified copy, no photocopies). In either case, you will need a guarantor for your visa -- usually your company/school can act as one for you; if they won't, you'll need someone (often they'll demand he be a Japanese man) with a stable job and salary who'll be one for you. Your new cute girlfriend or that cool dude you hang out with will not be acceptable.
The only other viable legal options to work in Japan are through a spouse visa by marrying a Japanese, or a dependent visa by being married to a foreigner in Japan who is legally working full time.
If you think you'll be travelling a lot around the country to look for work or go sightseeing, an excellent idea would be to get a Japan Rail Pass which allows you to go on any JR train in the country for one price paid up front. You must get the pass before going to Japan and it's also good for the bullet trains (except the fastest - the Nozomi). The price may look high but transportation in general is high in Japan.
Another important note: Once you're on a work or student visa, and you plan to leave Japan for travel, visiting family, etc. you MUST get a re-entry permit from the Immigration Dept. (take your passport to the local immigration office). A single re-entry permit is 3000 yen, and a multiple permit, good for an unlimited number of re-entries (for the term of your visa) is 6000 yen. IMPORTANT!!: Failure to get the re-entry permit stamp in your passport may prevent you from being allowed back into Japan.
Watch out for peak travel periods as well -- the end of the year, August, and "Golden Week" (the end of April and the first week of May) are usually jammed with passengers and prices are double or worse.