Advice/Recommendations

First of don't look for a hotel or Airbnb. Look for Hostels they are cheaper and pretty clean They also give you the chance to meet other travelers in the common areas where you can exchange travel tips or locations that they discovered to check out. As for the itinerary. If you are interested I can give you a nice one on Seoul based on the parts of the city. As well as Jeju I have only been to Busan once for a few days so I don't know much about there but I do know a few places in Haeundae.
Other than that I can recommend if it's your first time check out the big tourist attractions since they are really good places and make sure you go to Lotte World and Everland. They both have really different vibes and if you go in fall you might even be able to go during the Halloween season. A friend of mine works in Everland and they really make amazing costumes for Halloween each year so the parade is something to look forward to.
I would normally recommend a two-day trip one in Caribian Bay and one in Everland since they are close but during fall it is probably too cold to go to Caribian Bay so Everland would be a one-day trip from Seoul the Subway or Bus connections are really good.
For Jeju again more sure to look for a hostel or if you plan to travel around book hotels on the evening you are checking in so you have more freedom to go where you want to go. And for Jeju, if you have a driver's license you should get a car for the time there because the island has slightly more troublesome public transport. It is usable but with a car, you can reach a lot more places. Also, take the plane from Seoul Gimpo there and book either early or shortly before so the tickets are cheaper.
Jeju has a few interesting things like the Museum and Garden and a lot of labyrinths which are fun to explore but it depends on who you are going with.
Other than that my first tip would be to get a T-Money card at the airport for tourists and get yourself a pocket WiFi, not a sim card. The pocket WiFi can save your even though you have WiFi in a lot of places it is still important to use a translator or to look up directions.
Oh and MOST IMPORTANT Google Maps doesn't work in Korea so get yourself Kakao Maps or an offline navigation app to be able to look up where you have to go

One of my college friends grew up there and is there now. Remind me to ask him tomorrow morning

i think as long as you have water, it's a good place to be in

A place with a window would be good too

I would recommend a Unicorn floaty if you want to swim ♡

If I get another comment like these ones below I will jump off a bridge :knife:

Somewhere to sleep indoors would be nice

I think a place that has food will be good

Also I recommend you stay at a place that has a bed and a bathroom

wheres my invite