Nearly all visitors to South Korea visit Yeongjong without ever realizing it, as this is the island than Incheon's international airport was built on. The airport sits on a perfectly square-shaped bulge formed largely from reclaimed land, but the island is not artificial and its beaches, hot springs and fish markets make this an interesting way to spend a layover or escape Seoul for a day.
Incheon International Airport is located smack dab on the middle of Yeongjong, and is covered in detail in the Seoul article. Buses 223 and 203 connect the airport directly to Airport Town, and there are limited services to Yeongjong Wharf and the west coast beaches. As distances are short, taxis are also a viable option.
A local bus trundles from Seoul's Yeongdeungpo Station (Line 1) through Gimpo and Incheon Airports to Muuido and Eulwang Beach. At only W5000 this is the cheapest option, but the trip takes nearly two hours even in good traffic.
The most scenic option is to take a car ferry from Wolmido in Incheon to Yeongjong Pier at the southeast corner of the island. Ferries run every 30 minutes from 7 AM to 9:30 PM, the trip takes 20 minutes and costs just W1500.
There are also direct high-speed and regular ferries directly from Muuido to Incheon's Yeon-an Pier, but they only run three times a day (8 AM, 10:30 AM and 5 PM).
The 300-meter Yeongjong Grand Bridge links the island to the mainland.
Bus 202 runs around the south and west coasts of the island hourly, connecting most points of interest including Yeongjong Pier, Jamjindo and the beaches (W1200, hourly).
The primary attractions for most visitors are the beaches. Watch out for tides: at low tide, the waters pull back to reveal vast mud flats.
To get a little further off the beaten track, head to Jamjindo at the southwest corner of the island and take a ferry (W1000, roughly hourly) across to Muuido. Here you will find a fishing village, some cheap seafood restaurants, the quiet Hanagae Beach and the swimming pool-equipped Keunmuri Resort (W1000-2000, open Jun-Sep only).
Seafood is the thing to eat here. You can buy your own at the Yeongjong Pier market and have it cooked up in any of the restaurants nearby.
Airport Town Square has a series of hotels aimed squarely at overnighting travellers. Many offer free pick-up.