Ancient history is notable for the fact that a lot of research has always been done and is still being done in a range of national languages. Which languages are most relevant to you will depend on what topic you want to dig into. German was dominant for a long time and is still well represented on all topics, perhaps especially on the Ancient Near East. French scholarship is particularly strong in Greek social, cultural and religious history, but also in excavation reports from all over the Near East and Iran. Italian scholarship is inevitably crucial for those who want to study any topic within Etruscan and Roman history, since all the archaeological material is published first in the local language (and for the same reason, stuff on Greek archaeology makes knowledge of modern Greek a big plus).

If you want to go into historiography, German will be absolutely essential, since all the foundational works of ancient history are in German. The dominance of German scholarship in the early years of this field of study can't be overstated. When the first edition of the Cambridge Ancient History was published in the 1920s, its bibliography consisted of nearly 80% German titles. Anglophone scholarship now takes up the lion's share, but there is still a lot of research done elsewhere too, and Italian and French scholars in particular are rightly frustrated over the fact that a lot of British and American scholars can't or won't read their works.