Food in South Korea?

Hello everyone! I'll be traveling to South Korea in June and as I love to taste new foods and try out new things I'm really waiting forward to the trip! We'll be staying at the Seoul area and maybe a few days in Jeju. Any suggestions what foods I definately should try out (in addition to kimchi, of course!), good dishes or even restaurants? What about bento, is there any bento culture in Korea? Is there a possibility to buy bento boxes or other equipment? What foods or spices etc would make a good souvenir or I should maybe bring home with me? If you have any info or opinions I'd love to hear it! Thanks in advance :) T