Hello! That sounds like an exciting trip to South Korea! I'm happy to provide you with some travel tips.

Best Time to Visit:
The best time to visit Seoul, Busan, Jeju, and Daegu is in the spring (April to June) and autumn (September to November) months. During these seasons, the weather is mild and the countryside is in full bloom or beautifully colored with autumn foliage.

Route and Accessibility:
You can reach Seoul via the Incheon International Airport. From there, you can take the KTX high-speed train to Busan, and Jeju Island is accessible via a short flight or ferry from Busan. Daegu is also reachable by train from any of these cities.

Accommodation and Booking:
There are plenty of accommodations in these cities ranging from luxury hotels to budget guesthouses. It's advisable to book in advance especially during peak seasons. Check-in is usually at 2pm and check-out at 12pm.

Clothing and Essentials:
Pack a mixture of light and warm clothing, as the temperature can significantly differ from day to night. Don't forget a good pair of walking shoes for exploring the cities and hiking trails. An umbrella or raincoat might come in handy during the rainy season.

Health and Safety:
Tap water is safe to drink in South Korea. The healthcare facilities are excellent in these cities, though travel insurance is recommended. Follow common safety measures like watching your belongings and avoiding dark, isolated areas at night.

Local Etiquette and Respect:
Koreans bow when greeting, and it is polite to return the gesture. Avoid loud conversations in public. Tipping is not customary in South Korea. It's also important to respect the local customs and traditions.