Unlike most other developed countries, taxis are not a particularly expensive or "premium" form of transit in Korea. They are subsidized by the government, and actually feel more like a regular part of public transit. In fact, you can even pay for taxis by tapping your transit card.
Generally, taking a taxi in Korea is safe at any time. If you ask Korean people, there is still some level of distrust in taxis, as some people are worried about safety (especially for solo females) late at night. There are some unfortunate instances that have made big news in the past, but generally, Korea is a safe country and the chance of having a bad encounter is very low. The more common thing to be careful about is taxi drivers assuming you don't know the city and taking you on a slight detour to raise the meter. As such, it might be an idea to keep the route on hand (on Naver Maps or Kakao Maps) just in case.
In Seoul, most people tell the taxi driver to drive to a particular subway station. This may seem strange, but it's a very effective way to get roughly to where you want to go without the hassle of trying to convey an exact address. Of course, they can also input a specific place / address in their phone and drive you exactly there.
The main downside of taxis in Korea is that taxi drivers generally do not speak English. If you follow the method above, however, and prepare the subway station name in Korean, it should not be a problem. There are cases of taxi drivers refusing foreigners that don't speak Korean, but the best way to avoid worrying about this is to have a clear, concisely communicable destination they will understand.
While Uber itself no longer operates in Korea, there are similar apps in Korea, like KakaoT. These apps, however, are often hard to set up for foreigners without a Korean ID card or bank account. You can try, but it's probably easier to just flag one down in the street.
When flagging down a taxi in the street, you need to know which taxis are empty. Most taxis have this sign only in Korean, so you should distinguish between the following text:
Taxi Signs
Example of an Empty Taxi
Don't be confused by the green "예약" symbol, this does not mean the taxi is available.
Midnight to 2AM is one of the busiest times to catch a taxi, especially in areas known for nightlife. In this case, if you can't use an app like KakaoT, it may be hard to flag one down. It can make sense to walk to a nearby larger road instead.
After the ride, you can pay the fare by international credit card, cash, or by tapping your T-Money card.
