What did romans drink out of?

I believe they drank from goblets and flattened bowls.

Like many other things, the Romans adopted the Greeks knowhow and customs, and used clay dishes, often shaped in the traditional Greek forms, like these

They also used glass or metals, like these:

They also had some really expensive stuff, made of gold, ivory and semi precious or even precious stones, but I doubt they really used this stuff as daily kitchenware, and anyway only the filthy rich could afford them

The upper classes used silver one-handled cups and pitchers. The average Plebian class used terracotta or carved-stone two-handled cups and bowls. The Royalty used gold, but very little survives to this day, as this was likely the first booty looted and carted off by the Goths.