Hangul is the writing system used by the Korean language (both in south and north Korea) and by a couple of other minor languages. It is well known and largely appreciated for its simplicity and for its elegant design. You can find a very good introduction to Hangul at Wikipedia (this same page is available in many other languages):
Actually, Wikipedia goes far deeper in describing Hangul. The “See also” section of the mentioned page lists several other Wikipedia pages with more content:
It even supplies the curious reader with a page regarding the historical origin of Hangul:
And… a similar “wiki” project, called “Korean Wiki Project”, supplies a complete digital book (a “course”) devoted to Hangul and to the Korean language:
This last website, in particular, supplies a complete set of audio records of Korean consonant, vowels and dipthongs, played both by a human male native reader and a female one.
Another complete list of Hangul characters, with audio records, is available here:
And, of course, YouTube can literally flood you with an unbelievable amount of hi-quality videos:
So, it would make very little sense to replicate any information on these pages.
Rather, I have to say that none of these resources can actually be suggested as a “best” or “fast” way to learn Hangul. Nowadays, the best way to learn Hangul is a simple, nice, addictive application that you can install on your Android smartphone or tablet: “Write it! Korean”. You can find it in the “Google Play” store:
“Write It!” is a “game-based learning” app. It involves you in a game aimed to read, write, vocally recognize and speak hangul characters, dipthongs and syllables. It is composed by 18 different levels, each with a “practise” and a “test” section. Each section lead you in the process of drawing the character in the right stroke order, recognize the drawn letter and recognize the sound associated with each element. It is by far the most complete, the most addictive and the most effective app designed for this task.
So, do not wait longer! Just install this app and start playing. In a couple of days you will master the Hangul writing system and will be ready to start studying the Korean language.