Japanese is a unique and fascinating language, and thanks to innovative (free!) new learning resources on the web it is becoming easier and easier to learn on a casual basis. If you're like me and are keen to learn for fun, here's some simple steps to learning the different facets of Japanese from home.
Spoken First, Written Second
One of the most daunting things about learning Japanese initially is the concept of memorizing the complex writing system as well as the spoken words you are learning. The most practical outcome when starting to learn a new language is to be able to understand and converse in basic terms, so learning by Romaji (Japanese language written with English alphabet) to begin with is a good first step.
I had great success initially learning with Berlitz Essential Japanese, which I picked up from my local book shop. The entire book is focused on basic conversation and as such is taught entirely in Romaji. About.com Japanese is another good starting point for those who want to jump straight in and don't do books. By initially disregarding the written element you can get a feel for the structure and flow of Japanese which will make it seem more familiar. Once you have already learned some of the language in this way, it will be much easier to grasp the written side of things.
Spaced Repetition Software
Namely, Anki. This great little program is available for pretty much everything; Windows, Mac, Linux and a variety of phones (including your shiny iPhone) and helps you memorize a great deal of information quickly using a spaced repetition system.

As you learn new vocabulary, you add it to your Anki "deck" in a Q&A format (i.e. Q: Traffic A: Koutsuu.) Once the information is in your deck, you quiz yourself - the question is displayed on screen, and you think about what the answer is. You hit Enter to reveal the answer then rate how you did, with 0 being "I had absolutely no idea what it was" and 4 being "I recalled it instantly." According to your how well you fared, the card is then placed back into your deck, to be presented to you again further down the line at a time most opportune for your brain to commit it to memory.
By coming back to your deck every day and adding new information all the time, you ultimately commit everything you entered to memory, and very quickly. Anki has full support for Japanese characters as well. Cool!
Learning The Written Language: Hiragana & Katakana First
The first written Japanese you should learn is the phonetic characters, Hiragana and Katakana. Hiragana are the base set of characters for all the Japanese syllables, and used to write japanese words (i.e. さsa よyo なna らra - "Sayonara") Katakana comprises of exactly the same syllables, but written with different characters to represent words adapted from the English language (i.e. アa メme リri カka - "America")

These sets of characters are not difficult to learn. I recommend using Anki to commit these characters to memory quickly, as you can download pre-made decks designed specifically for learning Hiragana and Katakana. Once you have got Katakana down pat, you'll be amazed at how many English words you see in Japanese sentences!
Learning The Written Language: Kanji
Kanji are the more complex Japanese characters taken from Chinese. If you are learning for fun then it is probably not necessary to attempt to try and memorize the thousands of Kanji used in written Japanese. However, learning some of the more commonly used ones can be easily done over at speedanki. This free site is very similar to the Anki program in that it uses a spaced repetition algorithm, however it is web based.

SpeedAnki is based on the course content of the Japanese Language Proficiency Test, and there are four difficulty levels ("Kyu") of Kanji. The 4th Kyu is the easiest and contains the basic, more useful characters. You can continue learning Kanji in this fashion to any degree you like, and save your progress for later sessions. There are thousands of characters to be learnt through this site.
Immersion and Instinct
There are many pronunciations, nuances, idioms and exceptions in spoken Japanese, and the best way to become familiar with these is to expose yourself to the language regularly. Watching lots of subtitled Japanese film and anime is a good start and will help you to learn more simple vocabulary as well. Another good tip given to me recently is to consider the new words that you learn to be an addition to your native tongue, rather than a translation from a foreign language. For example, the Japanese word for delicious is oishii - you already have a bunch of words in your vocabulary for delicious; yummy, tasty, scrumptious, etc. Make oishii part of your vocabulary as if it were an English word, rather than a word you have to "compute" in order to say - that way it will come to you much more naturally.
Additional Resources
The Rosetta Stone - an innovative, although quite expensive, learning system in which no English is used, and instead uses images, audio and spaced repetition to teach a language natively.
iSpeak Japanese - a Japanese travel phrase book for your MP3 player, with additional vocabulary/phrase mini-book. I picked this up before I went to Japan to learn a few phrases before travelling. It was most helpful, and resulted in me wanting to study Japanese further. Having the audio also helps with understanding pronunciation.
JLPT Vocabulary Anki Deck - another pre-made deck for Anki with loads of JLPT course content including sentence structure and vocabulary. Some basic knowledge of Japanese as well as understanding of at least the Hiragana characters is required here. Knowing some basic Kanji will prove helpful too.
Denshi Jisho - Excellent online Japanese dictionary where you can search in English, Kana and Romaji.