First of all, Car rental is not a good choice. If you want to explore rural area and island in Korea then you may need to hire a car, else you do not need it. Korea, especially Seoul has very nice public transportation and heavy traffic. Don't bother yourself driving under the heavy traffic and make somebody else drive for you.
Then what is good public transportation for you? Here is my opinion.
#1 Uber is good but expensive. As the regulation and struggle against the government and taxi companies, Uber in Korea set their position as a super premium taxi. It will cost 2.5 times expensive than regular taxi. Korean taxi fare is relatively cheap, so 2.5 times expensive will not be super expensive but be aware that uber is luxury choice.
#2 Taxi is cheap, fast, plentiful so very easy to catch, but not very convenient for foreigners. Taxi drivers are usually old, and don't speak English. There are free interpretation phone call service in the taxi but I have never seen that service actually working, and drivers are not patient. And although drivers are usually honest but there are reports of overcharging foreigners.
One option is using "Kakao T " application, for iPhone and Android, which is Korean call taxi application. "Kakao" is Korean dominating SNS application corporation such as facebook, and Kakao T has English option. ( https://play.google.com/store/apps/details?id=com.kakao.taxi&hl=en ) This app provides similar service to Uber in the aspect of customer experience. You can call the taxi and set the destination without talking a single word to the driver. The navigation will show you expected distance and recommended route. Difference is that the price is flat (no surge!) and you have to pay in person : either cash or credit card.
#3. Metro is fast, cheap, covers almost 20 hours a day (05:00AM to 01:00AM), reach most area of Seoul, so it will be very good traveler's option. Price is relatively cheap. But in Seoul there are 9 lines, serveral hundred stations, so had better ready for transfer plan in advance. If you are not sure where you are going, then just ask any young people. Usually they can speak English, and will help you
#4. Bus system is also nice, but it is not very kind for foreigners.
My recommendation : #1. Use Kakao T taxi if you are not a super budget traveler.
#2, Ride on the metro first, and the ride on the bus for the last mile if needed.