Learning the two Japanese phonetic alphabets, hiragana and katakana, are key to learning basic Japanese. Each hiragana character represents a single vowel or consonant-vowel sound. In the chart below you can see all of the basic hiragana characters along with the closest sounding roman letters. The five vowel sounds, a (ah), i (ee), u (oo), e (eh), o (oh), are combined with the consonant sounds k, s, t, n, h, m, y, r, w to produce almost all the sounds represented by hiragana characters. The consonant-only n character appears at the end of words. These characters were all originally written with a brush, so writing the strokes of a hiragana character in the right order is important in getting the shape of the character correct.
The characters above are the ones you should teach yourself to read and write. Looking over the chart above you might be tempted to think "Why don't they just use Roman letters?!?" but it actually helps your pronunciation to learn to associate the sounds of Japanese words with hiragana. Your brain already associates sequences of Roman letters with sounds in English so it's important to leave that behind to pronounce Japanese words correctly.
Some of the basic hiragana characters above are also written with modifiers that change the consonant sound slightly. For example, by adding two small vertical dashes or a small circle, the pronunciation of the hiragana for "ha" becomes "ba" or "pa". The chart below contains the complete list of possible combinations:
When a ya, yu, or yo sound follows certain characters in Japanese, the sound is shortened and so instead of two separate characters, the sound is written as the combination of the first character and a small, subscripted version of the ya, yu, or yo sound that follows:
Once you've learned these characters, you're ready to start reading Japanese!