Taking bicycles on public transportation in South Korea is a fairly simple thing with only a few basic rules you need to know, so here they are.
Metro Systems: The rule for subways throughout the country is that bicycles are generally allowed on weekends and holidays. The front and rear train cars often have areas reserved for bicycles, though usually people will be sitting on the racks or seats that are supposed to be reserved. Additionally, there are some exceptions. In Seoul, for instance, Line 9 and the Sinbundang Line do not allow bicycles at any time.
However, if your bicycle is packed, even in a soft case like this, you can take it on the subway anytime you feel like it.
Inter City and Express Buses: Bicycles can fairly easily be stored in the luggage compartments in the undercarriage of these buses. A new rule for 2019 requires that bikes be packed in some kind of case or cover; however, on weekdays this rule is not enforced and you can put bikes in the luggage compartments without a cover/case. If the bus is full, be courteous and remove the front wheel to save some space. If the bus is empty, no worries. This is something I do regularly and never have an issue.
*I have not taken my bike on a bus during a weekend since the new rules were put in place. Because buses have more ridership on weekends, perhaps a cover/case is required.
ITX: The ITX train from Yongsan to Chuncheon is the most convenient Seoul area train for bicycles. Each ITX has 8 spots reserved for bicycles. These spots are generally easy to reserved on weekdays but sell out quickly on weekends so must be purchased in advance. Tickets can be purchased by the Korail Talk app: IOS here, Android here.
KTX: Bicycles must be in a case of some kind. Strictly enforced.
Intercity Trains: I have not taken a bike on any slow trains, but it is my understanding that regular trains require a case of some kind, though there are some trains with dedicated bicycle cars. That’s the best information I have.
Local Buses: Uncovered bicycles are strictly disallowed on local buses. Try to get a packed one on if you want, but these buses don’t have much space so good luck.
Taxis: This is up to driver discretion. But I can tell you from experience that taxi drivers do not like taking bicycles in their cars and typically require some persuasion to make it happen.