In order to adopt pets as part of your lance your character will need the Dominate Animal skill. Just walk up next to the creature you want to adopt and apply the skill to them. If you're successful then the creature will automatically become friendly and join your lance. A failure will result in the creature remaining hostile, and no additional attempts at dominating that particular creature can be made.
Pets count toward your lance's Leadership limit as per usual, so adopting more than one or two of them will require your character to cultivate their Leadership skill. If an animal is successfully dominated while the lance is full, it will not be part of the lance but will be friendly and offer the option to [Join].
Pets take care of their own training, just select 'do training' when they have got some XP saved up and they'll spend the points according to their own needs.
Almost every creature in the Federated Territories can be dominated, although some are much harder than others. Bonecracker the Albino Alligator is almost impossible to dominate, but makes an excellent pet if you can manage it.
One final point: pets only count as 'half a person' when tracking the limit for your lance size, so your character can recruit two pets for each person or robot they're short of their limit.