To be honest, I'd say you should try and do both at the same time. There's no point learning how to read and write if you never actually use it with other people in real life (well there is, but socially, there isn't). Similarly, speaking it is all fine, but what if you want to order from menus/stores? For me, I read and write in my little vocab/structure notebook but I also read what I write out loud. I also try and talk to myself in Korean and see how far I can get. I mean, I used to hate speaking out loud in French, but my grammar and writing was always fine on paper. I realized that I couldn't actually speak in French at the same rate that I could read it (although I can translate it at pretty much the same pace). So I've been trying to speak more with my teachers/friends and it's getting there. Case and point, develop them together.

I've taught Korean alphabets in Thailand before and was amazed how quickly students picked up the written language. I would say definitely learn spoken expressions first to get the feel of how the Korean language sounds. But it should be relatively easier than any other language to learn Korean alphabets. You can practice your listening skills by writing down what you heard in Korean and have someone check your spellings and etc.