The language of Amber is called Thari. It is spoken in Amber itself, and it is used as a trade language throughout the Golden Circle.
A shadow generally has one or more native languages. The Golden Circle is no exception to this; natives of the GC speak the language of Minos, Lyonesse, Kitezh, etc. ordinarily, but most speak Thari either functionally or fluently. Shadows beyond the GC may have some Thari-speakers if they have commerce with Amber; shadows that don't will not have Thari-speakers.
Fluency in Thari is a "freebie"; you can decide how much or how little Thari your character speaks, as suits your character concept.
Someone with a lore level of 2 in a place can speak a bit of the language (enough to manage basic daily living).
Someone with a lore level of 3 in a place can speak the language fluently.
Shadow travellers eventually pick up the knack of learning languages quickly. Those with the Shadow lore (UL-Shadow) at 2 or higher are assumed to be able to quickly learn languages with which they are not familiar (although this is certainly far from instant; this is explicitly a long-term learning process).
