I learned Korean on my own for a few months. I dropped it for a while because I was busy and then took a class at the local community college. I have learned a few things along the way. This is the order I would learn in: 1. Hangul - Make sure you learn some of the basic rules of the letters like how they sound at the beginning of a syllable versus the end (ㅅcould be sh or t sound for example). I missed the multiple sounds consonants make the first time I learned them. 2. Learn to type - I did this backwards and I regret it. I should have learned to type first. There are several web sites that track your WPM typing speed (I use 10fastfingers). 3. Take advantage of web sites like memrise to memorize basic prerequisites like numbers, telling time, etc. and vocabulary.

Hangul is one of the easiest alphabets to master given the fact that each consonant/vowel has only one sound. The pronunciation is the more difficult part for English/not native Korean speakers. That being said, I would suggest doing both. Learn Hangul but also start memorizing basic phrases such as 안녕하세요 (annyeonghasaeyeo) and 감사합니다 (kahmsahahmnidah). For the record, I make up my own romanization 'cause I can never remember the official rules. =)

응 ^

Definitely what Mojave said

Hangul first. Korean words are written in Hangul of course, so you can't learn vocabulary unless you learn the characters. Simple as that. Whatever you do, do not learn by Romanization. It will make learning more difficult in the long run. Hangul is really easy - it shouldn't take more than a few days; the sounds are more difficult - you can refer to Romanization if you must; install the Korean keyboard on your PC/smartphone so you can start typing right away - Korean uses the same keyboard as English, it just takes a few weeks to get used to where they are.