Arriving in Germany
Pet Passports
Upon arrival at your destination from a non-EU country, you are expected to have a passport issued by a registered vet. Any veterinary clinic can issue a pet passport: they will scan your pet's microchip and enter it along with your pet's information and your contact details in the new passport. In Germany, they often require you to redo the rabies vaccination at the same time (as this is the only vaccination required by law) so that it can be added to the passport.
Housing
The laws surrounding housing with pets tend to be quite murky and although landlords cannot outright ban pet ownership, they can require you to ask for permission. As a rule, you can expect to request permission from the landlord if you have a pet that does not fit into the category of "Kleintiere", basically small animals that don't make much noise (rodents, rabbits, reptiles, fish, small birds etc.). That means for dog, cat, and parrot owners, prior permission should be granted.
That said, it is not so easy to evict a tenant purely on the basis of pet ownership: there would need to be proof that this pet is disrupting neighbours or causing damage to the building. As you can imagine, forming a good relationship with your neighbours and informing them about any potential issues, such as your puppy making noise while still getting used to staying home alone, can go a long way. Nevertheless, one thing dog owners need to take note of is breed-specific legislation: while it is not possible to blanket ban dog ownership, a landlord can certainly forbid specific breeds and even dogs of a certain size.
Insurance
There are three types of pet insurance that you need to be aware of: liability, health, and operation coverage. Dog owners must cover the costs for any damage caused by their dog to property or other people, and liability insurance is mandatory for dog owners in Berlin. Health insurance is more for one's own piece of mind and is widely available for dogs, cats and horses. It can cover as much as wellness checks and post-surgery treatment, or as little as a minimal percentage of costs following diagnosis. Operation coverage is a type of health insurance that covers only operation costs. It is thus a more affordable option than full coverage, and may be a better choice depending on your circumstance.
That said, it is still very much a nascent market in Germany and therefore it is paramount to read the fine print on the insurance policies before signing. It is very common for insurances to stop accepting animals (even as young as 4 years old!) and to hike up prices to the extreme (typically from the age of 7 years). In addition, there are often limits on the coverage on a yearly basis or per diagnosis. If your pet has any pre-existing conditions or is a bit older, it is a better idea to set up a good savings plan as you likely will end up paying more for the insurance than you would have up front, or in the worst case: your pet may not be eligible for coverage at all.
Dog Tax and Registration
Germany is one of the last remaining countries that still levies a tax on dog ownership. Unfortunately, the funds do not go towards maintaining dog runs or other pet-related city expenses; it is a luxury tax to discourage people from owning dogs and it can be put towards whatever the government pleases. The exact tax amount differs from one region to the next: in Berlin it is €120 yearly for the first dog, and €180 for the second and subsequent dogs. So, for two dogs that equates to €300 yearly. You can register for dog tax by post or email with this form to your local Finanzamt. Dangerous dog breeds need to additionally be registered with the Ordnungsamt.
Since the 1st of January 2022, all dogs in Berlin have to additionally be registered with this central database. If you already have a dog, you have until July 2022 to register it, or risk paying a €10,000 fine.
Transportation
Many public transportation options in continental Europe, with the exception of intercity buses, allow for pets to accompany you. In Germany, all but intercity and international buses allow pets to accompany their owners. However, unlike in Scandinavia and some other countries, dogs travelling outside of a carrier need to have the equivalent of an unaccompanied child's ticket. As a rule, if travelling outside of a carrier they also need to be leashed and wear a muzzle. In Berlin, dogs are included on time-based tickets (day pass, weekly pass, monthly pass, yearly pass) - if you are travelling on a single trip, however, you will need to buy an additional ticket for your dog. In cars, pets are required to be inside of a carrier or to be otherwise secured.