We lived in South Korea for two years, but that was 10 years ago, so some of this info could be dated- double check everything. This website shows a lot of the places we visited on various weekend trips:
https://thisiskoreatours.com/suggested-korea-itineraries/. I would recommend that if you go to Jeju to go there first. Then you can hop over to Pusan or Daegu and see the southern part of Korea and head north to finish in Seoul. Jeju Air is like the Southwest airlines of Korea and fairly cheap to use to hop down there.
1- English speakers can be found in Seoul and many signs in the city are mostly in both languages. The city has a huge population of people from all over the world due to embassies, business, tech etc. so you could encounter a great variety of people on your trip. However, the moment you get to the outskirts of Seoul, the dual signing drops off. However, we found that 100% of the time we were out of the city, people went out of their way to help us because they were just as interested in us as we were them. We've had several times where strangers gave us rides in their personal cars and shared their snacks, off duty taxi drivers picked us up for no fare, and business people closed their shop to walk us to the location we were trying to find. The kindness knew no limits! Also, we found many fluent English speakers in the children and teens who were thrilled to use their English skills in real life situations. Koreans are incredibly kind people and demonstrating respect for their cultural values will get you further than perfect language skills. Avoid being loud, having confrontations or exhibiting frustration. Also, show respect to anyone older than you- offer seats, hold doors, stand aside. A few other cultural tips: if you have tattoos keep them covered up, women showing cleavage is inappropriate and no one shows bare feet unless at the pool. Lastly, use two hands when paying/handing over tickets, etc- give and receive everything with two hands.
2- The entire country of South Korea is almost exactly the same size (and nearly the same shape) as the state of Indiana, so you can easily move around and you don't have to go far to be "in the rural areas." High speed train was my preferred way to move north/south in the country, but if you sit in the quiet car understand it is the SILENT car. Intercity bus is how we got around in the northern half of the country. These are very nice, roomy and comfy full size buses where you can put luggage underneath and sit with a foot rest and look out the window as you ride along. You buy your ticket at the bus station before you board, but do not need an advance purchase. Buses go around the country with much frequency so you are not usually waiting more than an hour before the next departure. Depending on the length of the trip, the bus stops at a rest area with bathrooms and "food court" style eating areas.
**Food courts---- typically you go to a central cashier and select your food and pay for it. Then you take your receipt to the actual food stall and are given what you paid for. Food stalls typically have 3-D models or photos of their food, so take a photo of what you want and show it to the cashier and they'll get it rang up for you.
If you drive, which we did occasionally, do not plan to drive in any cities. (I did drive some in Seoul, but it's not for the faint of heart) Driving in the country is much easier and less congested, but you will be 100% GPS dependent. We experienced significant "lag" on the GPS, so often it would not tell us to be in a certain lane until we had passed the line of cars getting into that lane. It might be better now, but just anticipate frustration and delay. Also if you get into the rural areas (where it's local bus service only) know that busses can be infrequent... maybe once per hour... and plan accordingly.
3-Because we lived in Seoul, most of our travel was day trips from there (long days but always worth it.) However we did stay in hotels in Pusan, Daegu, and Jeongdongjin all of which have enough to do to fill multiple days.
4- Things to do: If you plan to shop a lot, do Seoul last so you can buy a cheap bag and bring home your purchases without having to lug them around the country. Insadong (traditional and artsy) and Myeongdong (youthful, modern, fun) are my favorite neighborhoods in Seoul- but you can stay anywhere and get around easily. Shopping markets are a huge deal. "Purse heaven" was a favorite as was "toy alley" but literally any kind of item you want to buy, there is a designated "market" where you go and all those items are there. Namdaemun market is probably the most famous and there is the night market in Myeongdong. Gwangjang Market is one of my favorite places to eat and is best in the morning up through lunch. Mung bean pancakes are probably my favorite Korean food, which are common here in this market. Also any kind of steamed buns stuffed with pumpkin, black beans or red beans is delicious. If you are at outdoor locations, ladies with carts will sell you handmade food items and this is inexpensive and safe to purchase. Usually they have kimbap (similar to California rolls). Other street food includes seafood "pancake" and my favorite Kimchi "pancake" or yaki mandu (fried dumplings.) With all major cities, there are amazing museums, cultural sites, parks, and lots of fun nightlife- just do what seems interesting to you. Hiking up to the Seoul Tower to watch sunset or sunrise across the city is a great way to get some exercise. Koreans love the beautiful outdoors and hiking is a very big past time. There are fantastic places to go on hikes that you can get to via the Seoul metro. In fact, the ends of the Seoul metro extend far into the country side and is a great way to get out for day trips without having to relocate lodging. If you are interested, there are public tours to the DMZ area but if you want to see the Joint Security Area (where the blue buildings are) be sure that is part of your trip plans with that tour operator. Very few providers include that area so you have to be clear. To make it easy on yourself, you might consider booking excursions with tour operators or hired drivers just to ensure you get to your destination without frustration and confusion. You have to decide how much of the day you are willing to spend on navigating public transportation to be "independent" versus just going on the tour and relaxing and enjoying yourself.
Korea is a safe country. You will not need to worry about physical safety there. I went all over the country by myself with a small child and never worried about a thing. I was a long-distance runner during the time I lived in Seoul and logged hundreds of miles in the dark, alone not once worried about safety.
May is a spectacular month to be in Korea with warm, mild weather and flowers blooming everywhere. The climate that month will be similar to DC/Virginia/Maryland. March and April experience "yellow dust" but it is typically cleared out by May, however you should bring/buy a face mask just in case it is still going when you first arrive.
One final thought- we did not go to Jeju. You have to fly there and we were interested in flying other places in Asia. It is popular with honeymooners and it is considered the "Hawaii of Korea." That phrase means it is a beautiful honeymoon/island vacation destination within country. That is where the similarities end. Friends who visited Jeju say it is pretty, but not at all like Hawaii, and the most interesting part was going to the "educational" museums about marriage intended for the honeymooners. Educational museums were a special subtype in Korea...I once drove out with friends to the “Mr. Toilet House Museum" and it was unforgettable! I was not the least bit surprised when the poop emoji took off because that was already huge in Korea
Have a great trip.
