Are there any surviving works from the ancient world describing how people learned foreign languages? If people learned a foreign language at that time, they most likely learned it through an immersion environment, but were there any courses or maybe some works that describe some tips? The most likely work of this type would probably come from the Ancient Romans, since many of the elites tried learning Ancient Greek.
Can someone point me to some works or maybe some current research in this area?
immersion? I'm not sure where you are getting that from. Most students in Rome would hire private tutors (including Greek) to learn Greek and formalized Latin. Here are some books on the topic Yun Lee Too- Education in Greek and Roman antiquity, J.F. Dobson- Ancient Education and Its Meaning to Us, Robin Barrow, Greek and Roman Education.
My Latin teacher at uni specialised among other things in researching ancient teaching materials. The materials for teaching Latin to mainly Greeks who wished to pursue business and law in the Roman Empire were called Colloquia, and bits and pieces have been preserved. They contain short texts about daily life (such as a visit to the public bathhouse, or taking out a loan). Presumably these would be read, translated, and memorised, just as you do with textbook examples nowadays.
If you want to see examples of these texts, see Dickey, E., ed. (2012) The Colloquia of the Hermeneumata Pseudodositheanaa: volume 1 Colloquia Monacensia-Einsidlensia, Leidense-Stephani, and Stephani. Cambridge Classical Texts and Commentaries. Cambridge University Press. ISBN 9781107020108
Also were there any materials to learn other foreign languages besides Greek or Latin?
You say the elites tried learning ancient Greek. This is true, the richer Romans either got a Greek slave that taught greek to them or they travelled/traded with the Greek.