Lions and hippos were used in the Colosseum but where did they get them from? Did they have zoos?
This may be somewhat of a disappointment, but they really did not get them any differently than anyone else who got exotic animals for the next couple thousand years. Animal trade is big business, always has been. Paid Roman soldiers or hunters could be sent out to capture or aid others in the capture of animals. Animals that Romans caught/helped catch however were typically of the less dangerous sort. The bulk of all animals, especially the dangerous ones, probably came from natives of whatever land the animal came from. They would be part of the exotic animal trade business.
Catching the animals was also not much different than you would expect, although perhaps much more dangerous at the time. Depending on the animal, different traps would work. Nets, pits, cage traps, etc. You can catch some animals better than others depending on which trap you use, as some animals do not fall so easily for certain tricks. So again, locals who knew the land and the creatures around them were best in these situations. The animals would be transported to Rome from that point, by land or by sea. Animal rights were not a thing until recent times so do not imagine these travel situations as being anything near humane. Unless the animal was worth a good deal of money and/or hard to find.
A zoo did not exist in Rome as we think of them. Some extravagant temples or palaces might try to recreate a scene from wildlife indoors by using different animals, plants, and trees, or just simply have a collection of different kinds of animals that that may or may not be open to the public