Sure, I'd be happy to guide you for your trip to Vejle, Randers, and Haderslev. They're fascinating places in Denmark with a lot to offer. Here are some tips for you:

Best Time to Visit:
The most favorable time to visit Denmark is from late spring to early autumn, i.e., May through September. These months offer the best weather conditions and longer daylight hours.

Route and Accessibility:
There are regular train and bus services connecting Vejle, Randers, and Haderslev. You can begin your journey from Vejle, proceed to Haderslev, and then finally to Randers. If you're driving, the roads are well maintained and clearly signposted.

Altitude and Acclimatization:
The altitude is not a concern in these cities as Denmark is a fairly flat country. Acclimatization will not be required.

Accommodation and Booking:
There's a good range of accommodation available in all three cities, including hotels, hostels, and guesthouses. It’s advisable to book your accommodation in advance, especially during the peak travel season.

Clothing and Essentials:
Denmark can be relatively cool and rainy even in summer, so pack clothing that can be layered. Also, include a raincoat, comfortable walking shoes, an umbrella, your camera, and don't forget your EU power adapter for your electronics.

Health and Safety:
Generally, Denmark is a safe country to travel with excellent hygiene standards and healthcare facilities. Nevertheless, it’s always a good idea to get travel insurance ahead of time. Follow local advisories and security guidelines, especially related to Covid-19.

Local Etiquette and Respect:
The Danish are known for their politeness and respect for personal space. Be punctual, dress moderately, and uphold the country's recycling and littering rules. When in public, keep your voice low - the Danish value their quiet time.

Enjoy your trip to Denmark and take back some splendid memories!