How to travel out of the country for the first time alone!?

I plan to travel to Seoul, South Korea next year from NYC in summer 2014. My question is, can i get some tip and advice as to how would i purchase the best price tickets to and from there from online or other sources? How long should i stay on average in your opinion? because i want to stay as long as a month! but i dont know if that is recommended for first time trip. How much money should i bring? What is there to keep attentive of? Like, warnings and what not to do or to do at another country? I never traveled before, barely even out of NYC so going alone to another country sounds terrifying, this is why i ask for personal experience and advice! i am currently trying to understand and speak korean as well to make it easier for myself when i get there, learn the culture and some history as well.

Also ive been checking on google for best and cheap flights from JFK airport to Incheon airport, Best price ive seen yet was nearly $1160.00. Would it be possible to find roundtrip tickets for even cheaper? Just curious.