You can choose your own specialty or roll on this table

Choose a non-magical creature to become your specialty. You have spent a great deal of time with this animal species, whether it be a dog, horse, owl, mouse, etc, they are what you primarily train and have taken an interest in as your passion. The special affinity you have for them is evident as local people recognize you from your services, awards, or honors and regard you as a resident expert. 

You get advantage on animal handling checks to befriend your Animal Specialty and their relationship with you may be strengthened even further through your talents. You can spend 1d4 days training/caring for them after which they will follow you and become a companion.

You're a benevolent owner, priding yourself on the well-being of your pets. Any animal that becomes a companion to you can receive first aid. You may declare that your Herbalism Kit is actually a first aid kit for animals rather than humans. You can use it to add your proficiency bonus to checks where you are looking for herbs for small first aid purposes and to stabilize a dying animal. 

*No animals can be commanded to attack or fight for you unless your class allows it. Furthermore, endangering animal companions through your training or commands may cause them to lose trust in you and decide to leave. 

Training your Companions:

During your downtime each day, you may work on various training with your companions. You can choose something on the table below or roll a d20 to determine the level of training you will work on. Next to each type of training is how many hours/days/weeks/months it takes to perfect EACH individual behavior (for instance, your dog would not be an expert at every single working command after 4 months but might be good at herding a small flock of sheep or tracking scents along short routes).

I have listed mostly items for dog training but you can substitute appropriate levels for training any animal (like problem solving puzzles of basic to extreme difficulty for crows or steps to carrying parcels for a homing pigeon).  If you can't figure out which category your tasks fit into, feel free to use your imagination. 

Once you train your animal in a task, it can reliably perform that task for you. You might help local herders get their sheep back with your dog, put on a show in the town square where your rat completes a maze, or ask your pig to search for truffles to sell, for example.