Hello! That sounds like an exciting trip. I'd be delighted to provide you with some travel tips for your visit to Budapest, Eger, and Pécs.

Best Time to Visit:
The best time to visit these cities in Hungary is between March and May or September to November. During these periods, you can enjoy pleasant weather and fewer crowds. December could also be a magical time if you enjoy Christmas markets.

Route and Accessibility:
Budapest, being the capital, is well connected by air with many international flights operating directly. From Budapest, Eger and Pécs are easily accessible by train or bus. The travel time varies between 2 to 3 hours depending on the chosen mode of transport.

Accommodation and Booking:
There's a wide range of accommodations available in all three cities from luxury hotels to cost-effective hostels and B&Bs. It’s recommended to book your accommodations in advance especially during peak travel season. Online platforms such as Booking.com or Airbnb may offer a wider range of options.

Clothing and Essentials:
Pack clothing according to the season of your visit. Don't forget comfortable walking shoes as these cities have several beautiful pedestrian zones and outdoor attractions. Sunscreen, sunglasses and a hat will be helpful during summer days. Also, a good city map or a mobile data plan to use digital maps would be very useful.

Health and Safety:
Hungary is generally safe for travellers. Be aware of your surroundings, especially in crowded tourist spots due to the possibility of pickpockets. Always ensure you have travel insurance before the trip, and remember to carry your essential medications.

Local Etiquette and Respect:
Hungarians appreciate polite manners. A firm handshake is the norm when meeting someone. Tipping is customary in restaurants and taxis, usually around 10% of the bill. Respecting cultural heritage sites and public properties is always appreciated.