More than 4,000 islands and islets surround the Korean peninsula. So, it's hard to pick just one to explore. But if you're mainly staying in Seoul and aching for an island visit, head over to the Incheon port and hop onto a ferry to Deokjeokdo; it's about 50 km from the coast.
This island has it all--recreational beaches; a thick forest of pine trees; and plenty of mountainous hiking trails.
En nautical route to Deokjeokdo, stand on the ferry deck and take in the sights of various islands and islets. Once you arrive, you'll be enchanted by the scent of pine needles and ocean air. In addition to a mid-day swim, don't forget to also take a walk on the sand during the low tides (later under the moonlight or in the morning after a late breakfast).