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The best place to start when learning written Japanese is with hiragana. This phonetic system is used for grammatic functions like showing verb tense and inflection, as well as for giving the reading for a kanji. Each symbol, or kana, represents either a vowel sound (a e i o and u) or a consonant and vowel sound (e.g., ma me mi mo mu). There is also a symbol for a nasal n. There are 46 symbols in all, plus markers to change unvoiced consanats to voiced ones (ta --> da) and h sounds to p sounds (ha --> pa).
The reason for starting here is that hiragana are the basic building blocks for Japanese. In fact, books for small children in Japan are written entirely in hiragana. Even in books and newspapers for adults, difficult kanji will have the reading written above them in hiragana, a practice known as furigana. They will also give you a great deal of insight in terms of pronunciation. Unlike English, Japanese is read exactly as it is written.
To learn more about hiragana and for practice exercises for writing and reading, see this article.
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One you have mastered the flowing symbols of hiragana, the next script you will need is the more angular katakana. Katakana encompasses exactly the same sounds as hiragana, so for every hiragana symbol, there is a katakana equivalent.
So what is the point of a completely redundant script? Well, katakana is simplified and much faster to write than hiragana. It's often used instead of kanji, for example, when Japanese put their name on a list. It can be written quickly and the reader won't have to wonder about the reading of the kanji. It is also used to add emphasis and for onomatopoeia.
More importantly for Japanese learners, Katakana is used for imported words. If you are an English speaker, many of these words will sound very familiar to you and you can add to your vocabulary effortlessly. After all, you don't need to check the dictionary for カフェー (kafee), バナナ (banana), or コンピュータゲーム (konpyuuta gemu).
To learn more about katakana and for practice exercises for writing and reading, see this article.
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Now that you've got the phonetic symbols down, it's time to tackle the hardest part: kanji. There are said to be about 50,000 kanji characters in Japanese, but don't panic, you only need a small fraction of those to be literate. The Japanese Ministry of Education has a list of about 2,000 characters called the jouyou kanji which are taught in elementary and secondary school. If you can master that list, you will be able to read and write well enough for daily life.
If 2,000 still seems like a daunting number, remember that many of them are very simple pictographs. For example, 山 (yama) or mountain, looks like its meaning. Many other evocative characters like 木 (ki/tree), 口 (kuchi/mouth) and 門 (mon/gate) are common radicals in other more complex kanji, so you can sometimes guess the meaning of a new character by the sum of its parts. For example, the verb 聞く (kiku/listen) combines the symbol for ear with the symbol for gate.
An excellent resource for learning the kanji which uses this pictoral approach is James Heisig's Remembering the Kanji. For an online Japanese-English dictionary that includes a fuction for searching for kanji by radicals, try the useful Denshi Jisho. And for more help on studying kanji and some free kanji flashcards, see this article.