Is there a way to gain an animal companion without taking levels in ranger, druid, etc?
Can someone just purchase an animal an train it?
You can buy an animal and train it with Handle Animal, yes. But it won't be an animal companion as a druid or ranger get one, so it won't increase with your level, you have to use your expensive actions to Handle or Push it, etc.
Cheapy's absolutely correct. You can have all sorts of trained animals and beasts with you, and even exotic creatures are possible. Another problem with this, though, in addition to those outlined above, is that by the time a character is capable of obtaining such a companion the creature itself is so far behind the power curve that it is ineffective in combat.
You might be able to overcome some of these issues with the Leadership Feat, but how it's applied and what type of cohort you are able to attract is going to have to be determined by your GM; there are basic rules for Leadership, but it's one of those Feats that everyone has an opinion on.
In the equipment section, under mounts and related gear, on the table is a listing for guard dog, but there is no description for it, like there is for riding dog. Perhaps a guard dog is a dog pre-trained with the "Guard" general purpose. If that is so, you have a pretty decent combatant animal for 25g. You would still need some ranks in handle animal, so you can roll to get a 10 to handle it, or a 12 to handle it when it is wounded.
Keep in mind, this is just and assumption based on the name of the animal, "Guard Dog", assuming that it means it has been trained to guard.
Also its still debateable if you can even take bloodlines that require the sorcerer Archetype Wildblooded with eldritch Heritage.