The exotic animals in the Rome were mostly bought from traders. Mostly traders from North Africa and the Middle East (for lions and tigers) and also from India and Persia (for elephents).
The Alexandrian Tariff, a document issued between AD 176-180 lists commodoties subject to taxes in Alexandria on the way to Rome. The document mentions Lions, Lionesses, Maneless lions, Arabian Onyx, and Cheetahs.
After the animals were shipped to Rome, they were delivered to the Beast-Master who trained them.
Sources:
The Alexandrian Tariff
"Animals for Show and Pleasure in Ancient Rome" by George Jennison