Hello! That sounds like an exciting trip to Hungary. Here are some tips to make your journey smoother.

Best Time to Visit:
The best time to visit these Hungarian cities is during May through September when the climate is warm and most festivals take place. Also, note that the shoulder seasons of spring (April to June) and autumn (September to October) are also pleasant with fewer crowds.

Route and Accessibility:
All these cities are well connected by Hungary’s rail and road network. You can catch a direct train from Budapest to Kecskemét, later proceed to Szeged by either train or bus, and finally, head to Debrecen via train. For intra-city travel, consider using local buses, trams and cabs.

Accommodation and Booking:
All of these cities have various accommodation options ranging from luxury hotels to budget hostels. Booking in advance via trusted online platforms like Booking.com or Airbnb is advisable to get the best deals.

Clothing and Essentials:
Pack summer clothes for your visit but remember to carry some warm clothing for cooler evenings. Don’t forget a good pair of shoes for city walking tours. Essentials also include travel documents, necessary medications, and power adapters for your electronics.

Health and Safety:
Hungary is generally safe for travelers, but, as always, keep an eye on your belongings, stay cautious in crowded places, and make sure to have travel insurance in place. Tap water is safe to drink across the country.

Local Etiquette and Respect:
Hungarians are typically courteous. Traditional greetings include a firm handshake. It’s also considered polite to say 'Szervusz' (Hello) or 'Köszönöm' (Thank you). At cultural or religious places, follow dress code standards and maintain a respectful behavior.