Just like that, how exactly was Latin taught in Western Europe? Which method was used? Which pronunciation?
Pronunciation of Latin in 17th century Europe
Although Erasmus a century earlier made an early attempt at reconstructing Classical pronunciation, it didn't catch on. Different national pronunciations were used in each country in the 17th century. This also applied to Catholic countries, and the Catholic Church in France for example, which used their own national pronunciations rather than the Italian national pronunciation (which much later came to be seen as the Ecclesiastical pronunciation).
Classical pronunciation didn't catch on in most places until post-1900.  In 1879, W. W. Story wrote in the North American Review: "In Germany, Latin becomes German [in pronunciation]; in France, French; in England, English." 
Ae was a monophthong in most places, undistinguished from e. The vowels of English Latin were appalling: the Great Vowel Shift wasn't complete until the end of the 17th century , but it's likely that already in the 17th century English pronunciation of Latin involved long e becoming /i:/ and long a becoming /eɪ/ or similar. French pronunciation of Latin was apparently considered by Erasmus to be the worst of all, involving "tempus" becoming "tampus" (cf. French "temps" /tɑ̃/), "qu" becoming /k/, "mihi" /miʃi/ and "u" being pronounced /y/ (the French "u" sound).  In England, "c" and "g" were soft (/s/ and /dʒ/) before front vowels; in France, /s/ and /ʒ/; in Italy, /tʃ/ and /dʒ/; in Germany, soft "c" either became /ts/ or remained hard /k/. 
Teaching of Latin in 17th century Europe
As suggested by user Alex B. in the comments, a recent book edited by E. P. Archibald et al., Learning Latin and Greek From Antiquity to the Present, gives some pointers (although not as many or as specific as we might like). The most relevant chapter is that of Prof. Robert Black, and concerns Italy in the 13th to 15th centuries - but he sometimes hints at (and other times explicitly refers to) things being much the same in the 17th century, although he rarely mentions to what extent they were the same outside Italy. 
"The first stage of the Italian medieval, Renaissance and early modern school curriculum consisted of learning to read: this skill was always acquired through the medium of the Latin language." He later states that this didn't change until the latter half of the 18th century.
Not all pupils progressed any further in Latin than learning basic literacy, but for those who did, "the next step was learning the parts of speech and memorizing the varying forms (morphology) of nouns, verbs, adjectives and participles" followed by reading elementary Latin texts and (only once elementary reading skills were acquired) beginning to learn to write. Eventually pupils progressed to writing letters, reading advanced (including Classical) texts (Cato was a favourite), and studying style and rhetoric.
The first textbooks began with the alphabet and concluded with syllables to sound out. This was followed by reading words and phrases. Once competent in the alphabet, pupils would learn to read the Psalms. Black states that the use of the Psalter to impart reading skills remained standard practice in Italy until the late 18th century.
Grammar was studied via the works of the (4th c. AD) grammarian Donatus, and the extensive use of Donatus continued at least in the 16th century.
-  https://en.wikipedia.org/wiki/Traditional_English_pronunciation_of_Latin
-  W. W. Story, The Pronunciation of the Latin Language, at https://www.jstor.org/stable/25100716
-  https://en.wikipedia.org/wiki/Great_Vowel_Shift
-  W. S. Allen, Vox Latina. Second edition, 1978, Cambridge University Press.
-  https://de.wikipedia.org/wiki/Deutsche_Aussprache_des_Lateinischen
-  R. Black, First steps in Latin, chapter in Archibald, Brockliss & Gnoza, Learning Latin and Greek from Antiquity to the Present, 2015, Cambridge University Press.