Travel
I recommend staying at an AirBnB or a hostel to save on cost. There are also international hotel brands such as Marriott and Hyatt that you can find.
Flight
Incheon International Airport
Seoul has two main airports, but for most international flights you'll find that they fly into ICN. There are airport limousine buses that depart every 20 minutes (which I find the most convenient going into downtown Seoul) or a subway line that connects the airport directly to Seoul. If you will be bringing large heavy bags, I'd recommend hopping on the limousine bus or taking a taxi. Again, I recommend the limousine bus since the stop announcements will be in English and they go to most places in Seoul.
For all the days along the way