This specifically addresses the pronunciation of words ending with -in or -ain. Let’s use vin (wine) as an example. In French class, I was taught that it was pronounced “ven” (with the nasal sound instead of the actual n being pronounced.) However, it seems that native French speakers pronounce it more like “vahn” (very similar to how they pronounce vend [sell] or vent [wind]). What is the “correct” way to pronounce this syllable?
I would think there may be regional variations. I do know that in Paris I ordered “vah(n)” in a cafe and the waiter took me for a Frenchman.
Québec French speaker here, and vin (wine) does not sound like “vend” or “vent”. I’m not sure I’d say it sounds like “ven” either; the sound is closer to the English word “vain”, without the n.
In general (or at least the examples I can think of offhand) words than end in -in and -ain sound quite similar…the difference is that the sound is a bit more drawn out in the -ain words. When saying words like poussin or câlin, the mouth gets pulled back a little at the corners, like when we say the word “inn” in English. For -ain words, like pain or bain, the sound is similar, but the mouth falls open more, like when we say the letter “a”.
At least that’s how I hear it!
ETA: the more I think of it, the more I think the accent might sound more like “vent” in France, but I can’t do a French accent to save my life, and the word is beginning to lose meaning with the amount of times I’ve just repeated it!
I think the sound of “in” in words such as “vin” is one of the key differentiators between Quebecois French and Parisian French. They are very different.
There’s a sound bit there that sounds alright to me.
I pronounce it more nasalized than that, but I grew up in the north of France, where it’s common to do that.
You’re right. I never thought about it before, but the more I sat here repeating “vin” and “vent” to myself, the more I began to think that both sounded right.
Then again, the OP didn’t specify if they wanted Québecois or Parisian French 
“vin” and “vent” are pronounced differently. There are 4 nasal vowels in standard French of France: “in” “un” “en/an” “on”. One of these (“un”) is well on the way to disappearing in much of France, so that “un tolérant” and “intolérant” are homonyms.
This is a bit simplified, but here goes:
If you want to pronounce the word “vin”, take the English word “van”, remove the final n, and nasalise the vowel. If you pronounce “van” in a typically American way, you will be saying “vin” more like a Canadian. If you pronounce “van” in a typically British way, you will be saying “vin” more like a French person.