It is worth noting that Airbnb may not be the most affordable option. Always check hotels and hostels (remember, they also have private rooms!) as well, to make sure that you know all the options.
For pets that are very old, sick, or otherwise affected by changes in their environment, travelling by land or sea can be a milder alternative to flying. When travelling by land or sea, you may give sedatives if deemed necessary and prescribed by your vet.
An up-to-date rabies vaccination is a worldwide requirement. If you plan on flying*, you will at the very least need a recently issued health certificate from a vet (usually within 5-10 days of flying).
Travelling with dogs on public transport in Germany has a sort of interesting bureaucratic charm to it. In most European countries, there is a fairly simple system: dogs either travel for free or there is a dog ticket for every situation.
Timing is of the essence when it comes to making preparations (vaccinations, rabies antibody tests, deworming, securing an appropriate transport carrier, booking a place for your pet on a flight etc.), so start research and planning at least six months ahead of time if coming from outside of the EU!