So, you want to train your pet? Let's start with the basics, with the expansion of time of legends there are now what are called pet slots.
Each pet now has slots with a unique cap 1-5. This means, if you tame something and it is 3/5 slots then you have the ability to train it up 2 more slots. If you tame a pet that is 1/5 slots you will be able to train it up 4 slots.