Hello! That sounds like an amazing trip. Here are some travel tips for your journey to Spain, primarily focusing on your first destination, Madrid.

Best Time to Visit:
The best time to visit these cities in Spain, especially Madrid, is during the spring (March to May) and the fall (September to November). During these periods, the weather is pleasant -- not too hot or too cold. However, these cities are beautiful throughout the year, so anytime is good, keeping in mind your convenience and preferences.

Route and Accessibility:
Madrid, being the capital of Spain is well connected by flights from all around the world. From Madrid, you can take the high-speed trains (AVE) to Valencia and Barcelona. Alternatively, these cities are also well connected by bus and flights.

Accommodation and Booking:
All three cities have a good range of accommodations to suit different budget types. I'd recommend booking in advance, especially during the peak travel periods. For the best deals, use reputed online booking platforms like Booking.com, Airbnb, or hotels' official websites.

Clothing and Essentials:
Lightweight clothing that can be layered is most appropriate. Carry a comfortable pair of walking shoes as these cities are best explored on foot. Sunglasses, sunscreen, and hats for sun protection are also recommended during spring and summer. Travel insurance and copies of your essential documents are a must.

Health and Safety:
Spain is generally a safe country with good healthcare facilities. Ensure that you have travel insurance that covers medical expenses. Hydrate frequently, especially during the summer months. Keep emergency numbers handy.

Local Etiquette and Respect:
Spaniards appreciate polite manners. A simple 'Hola' (Hello), 'Por favor' (Please), and 'Gracias' (Thank You) can go a long way. Respect local customs and traditions, especially during festivals.