Now that you and the person you are talking to have established the relationship and agreed on what form of speech you should use, the most important part of the conversation may start: what should you talk about?
Conversation topics vary greatly depending on personality, occupation, social class, age, gender etc. However, without knowledge of the Korean family structure, Korean education system or Korean entertainment you may encounter a tough time finding a topic you may share with Korean people.
Korean people seldom discuss their past lives. They are very future oriented, goal-driven people who see no use in discussing past events or memories. They may talk about past accomplishments but usually do so "modestly": they do not give every detail of how they made it to succeed.
Perhaps in order to know what to talk about with Koreans, it is better to ask them about what they do during their daily lives. Koreans tend to avoid specific questions like "what does your job consist in doing" or "how do you proceed in studying". They tend to speak more generally about what they do. Koreans tend to ask each other what their hobbies are and what their goals are in life in order to establish what they will talk about. "What are your hobbies" and "what are your future plans" are very common questions for which Koreans expect a clear answer.
Hobbies vary from person to person. Some people have many hobbies, while other may only have one. It is not considered deviant in Korean society to put all one's efforts into one hobby. Some will focus on learning English for an entire year, others on learning an instrument etc. While for those who have a great variety of hobbies it is very easy to focus the conversation on a great amount of topics, others may be ignorant of a great amount of topics. Middle class women, female students and housewives tend to spend a great deal of their time cooking and watching television and may never reject a conversation about Korean entertainment gossip or sharing recipes. Korean middle class men however spend a great amount of their time worrying about their bank accounts and are likely to discuss stock prices, the job market and studying English - which has become a crucial element to make money - more on that later.
As for future goals, both men and women may be busy preparing for very competitive examinations until their late 20s or early 30s. Many plan to go to graduate school abroad some day. Those married may plan to send their children to study abroad. Most study English to improve their TOEIC or TOEFL scores: a key to securing a promotion or good job position. Koreans readily discuss those topics, but do not discuss the details. While it's OK to ask them what they plan to major in, what they plan to study or how many hours they spend studying every day, most Koreans do not have discussions about the subjects that they will study. For example, they rarely discuss English grammar or what they plan to write their dissertation about. However, they do discuss what kind of job they expect, but do not discuss the logistic aspects of their future dream jobs.
Most importantly, Koreans do not like discussing topics which require specialization, no matter how close they may be to their friend. This means that politics, the economy, society or sports are topics that tend to be avoided. People may discuss their political affiliations with close friends, why they like or dislike the president, how interest rates have gone up etc. but tend to discuss those topics concretely, citing lots of concrete examples and very little theoretical data. Although Koreans do vocally disagree on certain issues with people they trust, many pretend to agree, or avoid topics that invite to discussion alltogether.
Finally, note that Koreans keep their explanations short but their descriptions long. This means the "how" rather than the "why" tends to be elaborated. For example, when complaining about something, Koreans will describe everything that happened, all the circumstances, but will not analyze what happened by issuing statments like "I think that when this happens he always does that in order to prevent yonder from happening". They prefer statements like
"after this happened he did that
_and did yonder happen?