Many expats come to Korea to replace another foreign employee. Usually, the Korean employer asks the leaving employee to write a letter or leave a note to the incoming employee, with bits of information about the job or neighborhood. However, this does not always happen.
Get your Home Address
Find out your home address, in both English and Korean. Have it written down in Korean, and keep it in your wallet. It's useful for getting home, and also getting things mailed to you.
Getting a Cell Phone Plan
You cannot sign up for a cell phone contract until you have been in Korea for three days. However, you can sign up for a prepaid plan geared towards tourists, and continue using that throughout your stay. At the airport, you can pick up EG Sim Card (29,900 won for 1 gb). Also, there is the KTisM SIM (19,500 won for 1 gb) from the 7-11 near the subway station.
Additional reading: Mobile Phone Networks
Apartment
Note that there are businesses that offer arrival kits for newbies, which include things like cleaning sprays and bedsheets. These kits are highly overpriced. For less than half the price of an arrival kit, you can buy the same things online, at local shops, and at Daiso.
Disposing of Trash
Trash must be separated into different categories. There are many different categories: general waste, food waste, hard plastic, soft plastic (vinyl), paper, glass, metal, cloth, batteries ... and probably a few more. You risk getting fined or yelled at if you are caught improperly disposing of your trash. For general waste, food waste, and construction waste (you are unlikely to use this), you must buy special bags from your city.
