It is hard to say which is easier because they are very closely interrelated, almost the same thing in a way.
Picking up 한글 alphabet is quite easy (half day) but that is just the first step of a long way to writing fluently.
Just uttering a few words would be easy but again that doesn't mean that you can carry on a conversation any time soon.
So the two are really not that different.
If you have had enough exposure to the language and built large enough vocabulary and grammar knowledge, both will become that much easier. You can't say which is easier - they are different manifestations of the same thing.
As James Cho said, learning Korean speaking without learning hangul is like learning English without learning alphabet.
I know there are people who learn better by listening, but from my experience, I suggest you to learn the alphabet and then learn pronunciation.
As Tony said, they're both related.
Of course, learning an alphabet/writing system is much less involved than learning an entire language, and it is also part of getting a strong foundation, so start with Hangul. Of course, in order to learn Hangul, you need to learn proper pronunciation.
And when learning a language, you ought to be learning both writing and speaking, along with reading and listening.
If you don't know Hangul, your practice will be improved very limitedly.
한글을 모르고 한국어 하는 것은 알파벳을 모르고 영어를 하는 것과 같아.
If you are surrounded by Korean-Speaking people. You can learn Korean without Knowledge of Hangle.
If not, Maybe you need to learn Hangul :)