Hello! It's exciting to hear that you're planning a trip to Seoul, Gyeongju, and Busan. Here are some useful travel tips for you.

Best Time to Visit:
The best time to visit South Korea, including Seoul, Gyeongju, and Busan, is during the spring (April to June) and fall (September to November) seasons. During these periods, the weather is the most pleasant with fewer tourists.

Route and Accessibility:
Seoul, being the capital of South Korea, is easily accessible via international flight. From Seoul, you can take the KTX (Korea Train Express) or intercity buses to reach Gyeongju and Busan. The cities are also connected by a comprehensive public transportation system of subways and buses.

Accommodation and Booking:
Accommodation options in these cities are plentiful, ranging from luxury hotels to budget hostels and traditional hanok stays. It's advisable to book your accommodations in advance, especially during peak travel seasons, through popular booking sites.

Clothing and Essentials:
Pack according to the season of your visit. If you're visiting during spring or fall, light layers with a jacket would suffice. For summer, bring light, breathable clothing and sun protection. Winter requires heavy, warm clothing with good quality winter gear. Don't forget a good pair of walking shoes, toiletries, power adapter, and any medication you might need.

Health and Safety:
South Korea has a high standard of health and safety. However, it's always prudent to have travel insurance. Make sure your routine vaccinations are up to date. For safety, always be aware of your surroundings, especially in crowded places.

Local Etiquette and Respect:
Respect for elders and authority figures is important in Korean culture. Never use informal language when addressing elders or superiors. It’s important to bow as a sign of respect. Also, be mindful not to disturb others when visiting religious or historic sites.