So why do Koreans take their shoes off inside? Well, apart from it reduces the amount of cleaning you need to do inside, to me the obvious answer is traditionally they would spend a lot of time sitting (eating or drinking) or laying (sleeping or resting) on the floor.
By the front door of someones home in Korea you will ALWAYS find a shoe shelf or cupboard where Koreans will keep their shoes. When I am visiting someones home I typically just kick my shoes off by the door and leave them on the floor, but if there is not much space or I think my shoes will be in the way I put them in the shoe cupboard.
You will also find shoe cupboards/shelves at the gym, restaurants, workplaces and schools.
As a subtitle cue for when you should take your shoes off, keep an eye out for a difference in floor height. The height difference may be as little as 2cm or as much as 30cm. Here is a picture of the entry to my apartment. You can see the tiled area where my shoes are (I am too lazy to put them in the cupboard) is lower than the normal floor level.
Restaurants - Not all restaurants require you to take your shoes off. Restaurants where you sit on a chair (I will call it western style) don't require you to take your shoes off. If the restaurant is a traditional restaurant where you sit on the floor at a low table then taking your shoes off and going bare foot is a must. Some restaurants have mixed seating arrangements (western style and traditional style where you sit on the floor). The two sections of the restaurant will normally be well defined and the traditional area will be on a raised floor (about 30cm higher than the western style area). Before stepping onto the raised floor, take your shoes off. Sometimes there will not be a cupboard so just leave your shoes where the floor steps up.
School/Workplace/Gym - If you work at the place or go there regularly you are expected to provide your own slippers for daily use. If you are a guest there will often be a set of communal slippers you can use. If not, you will need to ask to borrow someones slippers or go bare foot. Pictured below is the shoe cupboard for teachers at my school. Each teacher has their own cupboard with their name written on it where you can leave your shoes. Mine is in the very bottom right hand corner next to the clock.
Gazebos - Gazebos like the one pictured below are littered all over Korea. You should take your shoes off before stepping onto them. People will often sit on the floor and have picnics with their families and wearing shoes would be like standing on a picnic table.
Supermarket/Fast food takeaway/Department store/Movies/Hospital/Public Bathroom - You can leave your shoes on at all these places.