We are planning to visit South Korea in February to celebrate my husband's birthday. We particularly want to visit Seoul, Busan, and Jeju. Is 1 week enough to visit the country? What are the attractions we should visit and any tips on the hotels, transportation, and budget?
South Korea 2024
I've been to Seoul a few times but not the other cities. February could be pretty cold to do much - I've been there in January as a stopover and it was subzero, so we had quick bursts of exploring, peppered with visits to coffee shops to thaw out.
Seoul itself didn't have too much for the visitor - not compared to other cities in the region like Taipei or HK.
I've noticed you ask a lot of wide open questions; perhaps you could come back with more detail about what you like to do? That would help people give better advice. Are you looking for nightlife, nature, hiking, museums, architecture, beaches, city vibe, cuisine?
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I have never been into seoul, but it is one of my bucket list. And probably want to visit jeju to live my k-drama dream. 
Have fun in seoul!
One week is enough time for a fast-paced trip to experience the major attractions across South Korea's top cities of Seoul, Busan, and Jeju Island. In Seoul, spend a few days visiting sites like Gyeongbokgung Palace, Bukchon Hanok Village, Lotte World, and the busy Myeongdong shopping district. Stay in the Myeongdong or Itaewon neighborhoods, near the convenient subway system. To get to Busan, take the high-speed KTX bullet train. Once in Busan, dedicate two full days to explore Gamcheon Culture Village, Jagalchi Market, and the scenic Dongbaekseom Island, staying near the Gwangbokro subway station. From Busan, fly to Jeju City on Jeju Island. While most guides recommend 4-5 days to fully experience Jeju Island without feeling rushed, in your case three full days may suffice to enjoy the volcanic beaches, hiking trails of Hallasan National Park, and resort areas like Jungmun without extensive backtracking. Stay on Jeju Island also near Jeju City for transit access. Throughout your travels, make use of the T-Money transit card for subway, bus and ferry transportation. Be sure to pack for cold winter temperatures. Also be sure to sample classic Korean dishes like BBQ, tteokbokki, hotteok, soju and street snacks.
That would be a dream! I always dream to go to the destination where some of my favorite korean dramas been film with. I hope you enjoy south