Korea is a crazy country that can feel really overwhelming at first. Here is a condensed guide (or as condensed as I can make).
Wifi is everywhere so don't be afraid to travel where without a data plan.
Respecting elders is a huge thing in Korea. There are specific seats on the subway and buses (also reserved for disabled and pregnant women). DO NOT SIT IN THEM. Even if there are no people on the train/bus stay standing. You will get reprimanded for sitting in them. Also don't be surprised if you get strange looks for not giving up your normal seat to elders if those are full.
The subway/bus system (in Seoul and Busan) is super easy to navigate and cheap. Use it to save money. Full write up here.
There are weird buttons on the outside of most restaurants/shopping/hotel doors. You have to push to be let in. Don't be like me and stand in front of the door wondering why it isn't automatically opening.
There are clean public toilets EVERYWHERE. Like even on random hiking trails in Jeju.
Don't put chopsticks in standing up - it is bad luck and reserved for deceased guests.
Craft beer is having it's moment in Seoul - make sure you try a couple of them. They were surprisingly delicious.
Korea is known for poisonous spiders. Especially on Jeju island. Be careful.
Want to fit in? Koreans say "Kimchi" when they are taking a picture in place of "cheese".
Getting to Korea
The most frequented (and therefore cheapest) airport is Incheon Airport (code: ICN). It is brand new and is one of the biggest and busiest in the world. It also is kind of an attraction in it's own right with a skating rink, TWO movie theaters and a ton of shopping inside.
Getting around Korea
You can fly for very cheap around Korea on Air Busan, Air Seoul, Eastar Jet, Jeju Air, Jin Air and T'way Airlines. If you are like me and just plan as you go, even buying flights the day before once you arrive in Korea is still ridiculously cheap. (For example I flew from Seoul to Jeju for $50 and bought the flight the night before).
There is also a high speed train that connects all major cities. I recommend the train because it is more central in every city and also you can arrive just 20 minutes before to buy a ticket and board. The high speed train prices are sometimes exactly the same as the flight prices.
Once in the major cities the subway/bus system is very extensive and cheap. Even the small island of Jeju (the Hawaii of Korea) has a great bus system that connects all the major sites.
Full guide on using the subways here.
Which cities to visit
Obviously a must do if you come to Korea. It is the biggest city in with a population of 10 million. It can be compared to NYC - great culture, tons of food, huge subway connecting everything. Plus it has some of the most beautiful palaces in the country. Full Palace guide here.
Korea's second largest city and its major port. Huge subway/bus system to get around the huge city. I only spent 24 hours here but there is so much to do you could easily fill days if not weeks.
The Hawaii of Korea is absolutely stunning. It was named the top natural wonders of the world and for good reason. There is hiking, waterfalls, volcanos to explore and mountains to climb. I only spent 36 hours here but could have easily stayed for a week. Hotels and restuarants can be pricy. Also take the train not the ferry to get here.
I didn't get the chance to visit but it has some of the most amazing ruins. Most people say it looks like an open air museum
A major city located right outside of Seoul (and actually where you arrive if you fly through Incheon airport). They have their own culture and sites to see. The Grand park here is amazing especially during fall.
Another huge city between Busan and Gyeongju that I did not have the chance to visit. Heard it is great especially for a stopover.