There is actually a whole cookbook about this that generally forms the bulk of attempts to reconstruct Roman cuisine. There are other recipes scattered across the corpus of classical literature (Pliny the Elder and Cato being two notable examples), but attempts to reconstruct cooking, particularly the highest status of cooking, is going to rest on Apicius. It is worth while to jump around the link to get a sense of what Roman "feast dishes" might be.

But to directly answer your question: Pretty much anything you can think of would be available, provided it came from the Old World. You want roasted parrot? It's in the book. Ostrich? There is a whole section on them. You want to make a scrumptious giraffe neck? It's been found in Pompeii. There is a real element of conspicuous consumption in elite Roman dining, in which exotic ingredients would be prized for their exoticness.

As for how you get them, there were some ingredients, like black pepper from India, that were so common that you could simply buy them in shops. But it is a bit harder to know how that giraffe leg ended up in a Pompeii midden. Mosaics from North Africa suggest that hunting exotic animals for export was something of an industry there, but what happened after the ship landed in Italy is a bit more difficult to suss out.

And a follow up, do we actually know of any famous roman chefs?