Sure, I would love to assist you. Here are some travel tips when planning to visit Eger, Tokaj, Villány :

Best Time to Visit:
The best time to visit these regions in Hungary is from May to September when the temperature is mild and pleasant, perfect for wine tourism and sightseeing.

Route and Accessibility:
Eger, Tokaj, and Villány can be accessible by trains and buses from Budapest. The travel time may vary for each city. Eger is roughly 2 hours away, Tokaj is about 3 hours, and Villány might take up to 4.5 hours from Budapest by train.

Altitude and Acclimatization:
These regions are all located at low altitudes, so acclimatization should not be an issue for visitors.

Accommodation and Booking:
All these regions have a broad mix of accommodations ranging from luxury hotels to guest houses. It is highly recommended to book in advance, especially during peak seasons.

Clothing and Essentials:
Pack casual and comfortable clothing for this trip. As these cities are famous for wine tasting, sturdy shoes could be useful for visiting vineyards. Don't forget to pack a camera to capture the beautiful landscapes!

Health and Safety:
There are no specific health risks in these regions. However, personal safety measures like not walking alone at night and keeping personal belongings close should always be maintained. Also, remember to consume alcohol responsibly during your wine tours.

Local Etiquette and Respect:
Hungarians are generally friendly and open. When visiting wineries, give due respect to the rules and local culture. Remember to tip in restaurants and cafes.

In Tokaj, you can be part of the famous wine auctions and participate in the Harvest Festival in September-October. In Villány, you can visit the historical wine cellars in the Villány-Siklós Wine Route.