I'll answer for the late Middle Ages, since that's my jam.

Multilinguality, or being a polyglot was the default setting during the late medieval period - single language societies were the odd ones out. Naturally, there are many ways in which you could speak several languages, from barely mutual intelligibility to full fluency in a foreign language. An example is French in medieval Scotland, which is used in war and diplomacy with foreign powers, but there is nothing to suggest it was ever used among the Scots themselves. Moreover, languages themselves were not beholden to a standard like they are today. Even though a border may have been drawn between say, France and Flanders, people in neighbouring villages would have been able to understand each other.

As for the actual learning of a new language, it is not so different from today: through speech and communication you can get far enough, think for instance of the lingua franca of the merchants. Now, if you are literate, there are written grammars available in Latin, French, and many other languages. Especially during the later middle ages, you see an increase in these treatises being produced.

(It's old, but a classic!) Bernhard Bischoff, 'The Study of Foreign Languages in the Middle Ages', Speculum, (36.2), 209-224. (A nice case study on how to acquire a second language) John Hirsch, 'Latin Prayers as a Lesson in Writing and Devotion for a Lady of Standing', The Chaucer Review, (41.41), 445-454. (A bit heavy on the theory, but absolutely fascinating!) Mark Amsler, Affective Literacies: Writing and Multilingualism in the Late Middle Ages, (Brepols: 2011)