Like many things, the notion of friendship differs from country to country. In many Middle Eastern countries, people consider themselves "friends" the minute they meet, in some European countries, continuous contact is required in order to maintain friendship, in the United States, both distances and caring are necessary for two people to be considered friends.
As I mentioned before, Koreans place high value on trust and do not trust people unless they are affiliated in some way. "Affiliation" differs from person to person in Korea. Some people require that their friends belong to the same big organizations: company, school, church etc. Others consider that smaller organizations like clubs, cafes or housing can be considered as a common affiliation. Others consider that a friend's friend can be a friend, while others reject this idea.
After introducing each other, Koreans tend to be quick to exchange their phone number. However, it usually takes time before they actually call each other, and if they call each other relatively shortly after meeting, the person who receives the call may feel suspicious.
In schools or companies, Koreans will build friendships by having common luncheons or drinking parties. Koreans seldom have lunch face to face at the beginning, as they prefer social gatherings where a group of people is involved, and will select their friends from that group.
Koreans don't feel the need to call their friends continuously to maintain the friendship or to meet everyday, they only call each other or meet when they have very specific purposes. Calling "just to say hello" is not common in Korean culture.
Koreans call their friends when they have specific needs: they may ask their friends to help them find a significant other, for advice regarding jobs, or for information regarding the organization or when they need any other favor. Koreans may be in permanent contact when such issues require long actions. They will call each other every day to see how a situation progresses. It is not considered rude to call several times a day to see whether the person responded to the favor, and to remind the person to take quick action.
Korean friends of the same sex may walk hand in hand (women) or arm in arm (men), touch each other's hair (women) and compliment each other on their beauty (women). Men however do not appreciate compliments on their beauty, but do not mind compliments on specific attributes like their eyes, skin or weight, as long as those compliments are from a person of the opposite sex.
Men and women tend to be highly suspicious of their intentions when they are friends. Men seldom meet women face to face with no specific purpose. Often, women will bring another friend along when meeting a man.
Finally, in big cities, it is not uncommon for friends not to meet for months, if not years, despite living in the same cities. Koreans are often busy with work, family or preparing for competitive exams to get a job and don't view meeting friends as a necessity. Friends are in no way offended by this and will help their friends out when in need.
Note that it is acceptable for friends to ask their friends to pay if they give them a favor requiring long work, such as writing a document, proofreading something, teaching, and in some cases, when the favor involves traveling. In that latter case, friends will give pay for their friend's transportation, and will give them extra money. For other favors, the tradition is that the person who receives the favor buys lunch or dinner to the person who gives the favor.