Learn Croatian: Bankomat mi je dao samo velike novčanice. - The ATM gave me only large banknotes.

In Croatian, enclitics (short words with no accent) follow a strict order. Usually, auxiliary verbs come before pronouns. However, the third-person singular auxiliary je is the big exception! It always drops to the very end of the clitic chain. Therefore, the dative pronoun mi comes first, resulting in mi je.

What case is velike novčanice here, and why?

It is in the accusative case, specifically the feminine plural. The verb dati (to give) takes a direct object in the accusative (the thing being given) and an indirect object in the dative (who receives it, which is mi here). Because novčanica is a feminine noun ending in -a, its accusative plural form ends in -e.

If novčanice means banknotes, how do I refer to coins or small change?

Coins are called kovanice. If you want to talk about small change collectively, you can use the masculine noun sitniš. For example, if the machine only gave you change, you would say Bankomat mi je dao samo sitniš.

Could I place samo at the very beginning of the sentence?

You could, but it would change the meaning. If you say Samo mi je bankomat dao velike novčanice, it implies that only the ATM gave you large banknotes (and the bank teller, for instance, did not). Keeping samo right before velike novčanice correctly focuses the emphasis on the banknotes themselves.