Absolutely not. I learned it before I learned Hangul, and I wish I never had. For some of the sounds, you really can't express them well with Romanization, so it doesn't make longer, more complex words easier to sound out. Also, My English speaking eyes are drawn to Romanization when it is present, away from Hangul. Hangul is so easy to learn, there really is no point in trying to learn how the pronunciation is supposed to work with Romanization. Set aside a few hours, find some Hangul flash cards online and a Korean friend to help with pronunciation, and you be well on your way to never needing anything BUT the Hangul. Best of luck to you.

Would you recommend a Korean to learn English using the Korean characters? Latin characters can only "come close" to what Korean letters represent in terms of phonemes, so you will get accuracy in your pronunciation if you learn the Korean characters from the beginning, if you try to go from Latin characters to Korean, you will have a feeling of uncertainty about how to pronounce words in Korean... aside from that, reading Korean characters will help you discover the prefix and suffix patterns a lot faster than if you try to learn them through Latin characters, since it usually the case that one Latin letter cannot represent a specific sound in another language and thus you have to form "clusters" of Latin letters in order to represent a sound that in Korean has a specific character. I hope this helps you.

you can try this for the start - learn-to-read-korean-in-15-minutes as for how romanization can be useful or not.. umm, it was quite useful for me AFTER I've leaned most of Hangul - I could check myself in case I doubted whether I've read the Hangul character(s) correctly. like when choosing between the two ways how the padchim should be read in this particular case - ㅅ or ㄷ. but again, it was when I already knew how those ㅅ and ㄷ should sound in Korean and when I felt very comfortable having only Hangul before my eyes - not earlier. thus, I totally agree that one should learn Hangul first - in order not to have his/her eyes fixed on the Latin (or Cyrillic) letters instead of Korean ones ;)

I wouldn't recommend using romanization alone, but when I was learning English, sometimes, I needed to write how to pronounce a word in Hangul next to the words. I don't agree entirely with @Mojaves. Hangul characters usually take longer than "a day or two". In fact, because Korean is the only language that uses Hangul characters, it may take a few days to a few weeks to learn ALL the Hangul characters and pronunciations for each one.

I agree with everything said below, but I'll add that if you are serious about learning Korean, then you need to learn Hangul. If all you want to do is sing along with songs, than Romanization may be enough. But to actually learn the language, the grammar, and be able to read and write, learn Hangul now. As mentioned, it takes only a day or two. Also, learn how to type in Hangul. Korean uses the same keyboard as English, just get a printout of where the letters are and you can pick it up quickly.