Offer your new landlord the opportunity to visit you after you've moved in so they can meet your pet and see how well you keep your current rental unit.
Try to take a few days off when you move into a new place to help your pet adjust. It's new for your pet too, and sometimes even the most quiet and calm pets will get anxious in new surroundings and make excessive noise, disturbing the neighbors. It often helps if you can be there to help your pet adjust to the new home.
Be a good neighbor. Make sure your pets don't disturb your neighbors, whether it's with noise, wandering loose, or unsightly messes. Remember that your landlord has to deal with complaints and won't be happy if it keeps happening!
Be diligent about addressing any concerns your landlord may have. If an issue arises about your pet, make sure you understand what the problem is and take immediate steps to address it. For example, your dog may bark excessively when you first move in as a result of being unsure of the new surroundings. Try another temporary solution (put your dog in a comfy covered crate with bedding, toys, and water; take your dog to a doggy daycare; take a few days off to help your dog adjust...)