Two bases for you: Jeju-si (city) or Seogwipo town where the Jungmun tourist complex and high-end hotels are. The difference between high-end hotels like Lotte in Jungmun and cheaper local hotels is not just the price but also how local hotels are run: you don’t get big bath towels or disposable bedroom slippers, and toiletries in large bottles are used successively by all guests.
Exchange rates are better in Korea than in your home country. Change your currency at the Jeju International airport, or withdraw cash from the banks that accept your credit cards.
Basically, two categories of things to do: all the cutesy museums like Hello Kitty and Teddy Bear museum; or the natural attractions of volcanic craters, parks, rugged shorelines and waterfalls. Don't count on huge shopping complexes. Jeju city has the underground Jungang shopping mall but Seogwipo hardly has any fashionable shopping district.
As taxi fares can be quite expensive, do stay near to the inter-city bus terminal in Jeju City or near the stops of airport bus no. 600 (usually big hotels). This bus runs via the west coast from Jeju City to Seogwipo and it can get you to some of the attractions with a short taxi ride away. The bus stops at every classy hotel in the Jungmun tourist complex in Seogwipo, and can bring you to the e-Mart and Olle market.
About museums, select what you really want to see. For example, the Chocolate Museum does not have useful information about the goodness and history of chocolates, only, a display of various chocs from around the world. The most enjoyable museum would be the Alive Museum where you can pose interactively and humourously with large paintings.
If you love seafood, you can eat a lot of fresh baby abalones in spicy kimchi soup. Abalones are farmed there. Bimbimbap is the national staple dish and white noodles with pork slices is their preferred breakfast fare.
With tangerine farms, edible seaweed thriving on the clean beaches, and temperate weather, Jeju makes for a lovely, peaceful and safe holiday. To top it all, do book online buy a ticket to watch the Nanta show in Jeju city. It’s a very funny mime of four cooks preparing real food and making music with their choppers and knives, and of course having their little quarrels and jealousies!