The basics of the fare system is actually quite sane: you pay a certain “base fare” whenever you first tap on, which covers up to a certain distance, after which you have to pay an additional “distance-based fare” for each certain additional distance traveled.

Why am I being so vague, you ask? It’s because all of those values are different for the exact mode and route you get on. For almost all rail lines, the base fare is 1550 KRW and covers 10 km, and afterwards you pay 100 KRW for each 5 km traveled up to 50 km, after which you pay 100 KRW for each 8 km traveled. For Seoul buses, you pay either 1200, 1400, or 1500 depending on what kind of bus you get on, which covers your entire trip unless you transfer to something else, from which point your base fare now only covers the first 10 km and then you pay 100 KRW for every 5 km traveled, except if you get on a night bus or express bus, in which case your base fare is either 2500 or 3000 KRW and covers up to 30 km.

Not so bad, is isn't it? But this is just the basics, and it merely covers most buses and railways that run within city limits.

For certain rail lines, when you transfer to/from them you have to pay an additional transfer fee on top of the base rail fare and the distance-based fare. U-Line in Uijeongbu will charge a flat fare of 200 KRW when you transfer between it and Line 1. The Sinbundang Line is notorious for having an absurd fare structure, where the line is split into three sections based on which private partner funded the construction. The northernmost section between Sinsa and Gangnam will charge you 700 KRW, the middle section between Gangnam and Jeongja will charge you 1000 KRW, and the southernmost section between Jeongja and Gwanggyo-Jungang will charge you 1000 KRW, except if you use the middle and southernmost sections together the two will only charge you 1500 KRW combined. Because the fares are based on sections, rather than distance, you can be charged upward of 1000 KRW just because you went a little bit beyond the section boundaries. The northernmost section, in particular, is notorious for this, as the section boundary between the northernmost and middle section is at Gangnam station, in the middle of downtown Gangnam. If you don’t know that that’s the boundary between the two sections, you can suddenly be charged an additional 1000 KRW on top of your base fare while just travelling within Gangnam.

As another example, the airport railroad, which is Seoul’s only airport rail link to Incheon Airport, has an exception carved out in the fare system only for the section of the line that is not on the mainland. As soon as you leave the last station on the mainland, you are charged a completely separate base fare of 1200 KRW which covers 10 km, and a completely separate distance-based fare of 130 KRW for every. single. kilometer. traveled.
Not only is that ludicrously expensive and needlessly complicated for the first line that most foreigners visiting the city are going to use, it actually creates a weird situation on the last mainland station, Cheongna Int’l City, where upon entry you will be charged the mainland fare of 1550 KRW, and if you go less than 10 km onto the island, the correct fare for you is only 1200 KRW, so the station staff have to come out to refund you 350 KRW.

There is also a weird upcharge clause if you cross province boundaries between Gyeonggi-do and Gangwon State or Chungcheongnam-do, where if you cross these borders, your trip will be split into two sections at the border. The shorter section will recalculated to a distance fare rate of 100 KRW for every 4 km, while the longer section remains at the 100 KRW per 5 km with the discounts past 50 km. This effectively means that crossing the provincial boundaries on the metro is really expensive.

Oh, the buses. Gyeonggi buses are basically the wild west. The base fare is different depending on bus type and sometimes depending on the city. The “town buses” are run by individual cities within the province and charge a different base fare depending on which city operates the bus. The “regular buses” have a uniform fare province-wide, except only some routes will let you cover the entire trip on the base fare of 1650 KRW, while others will only let you cover 10 km on 1650 KRW. And if you ever need to get on a red-painted long-distance express bus, the base fare of 3200 KRW will cover the entire trip, except if the bus is an orbital route that doesn’t go into Seoul, in which case the base fare will be 3450 KRW and only cover 10 km.

If you ever find yourself on the outskirts of the metropolis in Cheonan, be assured that Cheonan Buses do offer free transfer to the Capital Area Unified Fare System, except only when you’re transferring to/from Line 1, and only if you transfer at one of the 12 designated stations. Transferring to a Cheonan bus works completely differently to other buses, in that Cheonan buses do not work on a base fare + distance fare system.
Instead, you will be charged the base fare of Cheonan buses or Line 1 (whichever is higher), which will cover all of your segments on the Cheonan bus system and 10 km on the railway, and the distance-based fare of the Capital Area Unified Fare System for more than 10 km you have travelled on the Unified Fare System. You may be charged an additional city limits fare if you take the Cheonan bus beyond Cheonan city limits, except if you take the Cheonan bus to/from Pyeongtaek or Asan.
You can chain transfers to transfer from another bus route or other rail lines to the Cheonan bus, but you must have taken Line 1 and gotten off at one of the 12 designated bus stops immediately before getting onto a Cheonan bus. You can then chain transfers from a Cheonan bus onto a bus route run by Asan, but not directly from Line 1 to an Asan bus route, even though Line 1 literally terminates in Asan.

This system also has weird edge cases that let you pay less by riding more modes of transportation. There are two plausible scenarios where this can happen, one is when you try to transfer from a bus run by Pyeongtaek to a bus run by Cheonan, where you will pay less if you chain metro Line 1 in between the two bus routes. Same goes when you try to transfer from Line 1 to a bus run by Asan, you can pay less by chaining a Cheonan bus between the two bus routes.