Fun fact: I went to middle school on jeju island!
One of the most asked questions when people visited us at mija seoul was whether it's worth it to go to jeju island. And my answer to that question is a definite yes as long as you know the right places to go + you can drive / going with someone you can drive.
Transportation on jeju island is a little tough without renting a car since there's a mountain in the center of the island so there's no direct path from north to south. But if you're going for more than 3+ days, no worries as you can take the island by sections.
My suggestion is to book a hotel in jeju city and seogwipo and cover both the east and west coast.

These are all the places we went to if you want to use them for reference!
BadangGil
https://maps.app.goo.gl/Wu9XpZ8SDgSmD...
Hyeopjae Beach
https://maps.app.goo.gl/nZ6L8TYaK4Gdg...
Cacti in Wollyeongri
https://maps.app.goo.gl/r4h9gVPk66XpN...
Aewol Matcha
https://maps.app.goo.gl/ZMu6RMxs3c3Du...
GYORAE Black Pork Main Branch
https://maps.app.goo.gl/SbF41Yv4RAUKL...
Mulyeongari Oreum
https://maps.app.goo.gl/DSavdw31eGSPp...
Seogwipo Olle Market
https://maps.app.goo.gl/BBWv3g2hyvGAs...
Olle Jangtuh Gooksu
https://maps.app.goo.gl/YEpHdssCv4uXr...
Jeil Rice Cake Shop
https://maps.app.goo.gl/RbezzKs1sCyCt...
해녀의 집 (산이수동항)
https://maps.app.goo.gl/GsqTpWenFEcXB...
Miyoung's Sashimi
https://maps.app.goo.gl/CtdGq6rRXQ9n3...
Doldam Carrot
https://maps.app.goo.gl/Qpa4x6CyCm8BV...
Kalchi Wang Sungsan Location
https://maps.app.goo.gl/Ssh189RUpyGpq...
Gelateria_SUM
https://maps.app.goo.gl/ScpnAx2Fhse1q...
성게애빠져
https://maps.app.goo.gl/1MReAKR7psaHG...

To watch Haenyeo Women Divers, make sure to go to Seongsan Ilchulbong Peak at 1:30pm and 3:00pm. It's held daily!

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