Now, We went through all the vowels in English, but what about the ones not on the list.  We will not go through them and see how many of these can apply to the languages you study.
Well, we know about the front-high vowel /i/ and /ɪ/ but now we have the same thing only rounded.  Rounding has to do with how your lips are shaped.  The rounded versions are /y/ and /ʏ/.  Some might actually produce this in English word <dude> but not all of them.  This is the French <u> and the German, Turkish, Finnish, Hungarian, and Pinyin <ü>.  It is also the Norwegian and Danish <y> (from which the symbol came).  This also means that to make this sound all you need to do, is say <ee> with rounded lips.  This of course brings in the semivowel /ɥ/ which is like saying /j/ with rounded lips.
Now below that we have /e/ and /ɛ/ which have the rounded forms of /ø/ and /œ/.  They are the <ö> of German, and most of the above mentioned languages.  This sound is also in French under different circumstances.  This is like saying <ay> with rounded lips.
Below that we have /æ/ which is rounded into /ɶ/.  I rarely have seen this, but it's basically /æ/ with rounded lips.

The we have /ɨ/ and /ʉ/.  The first one is the unrounded form of a high central vowel.  This is like /ə/only higher.  This is in Russian, and most of the Slavic languages as well as many others.
Then we have /ə/ and /ɵ/.  We know /ə/ as like the <uh> we have.  If you say that <uh> with rounded lips you'll get /ɵ/.  I've never encountered that vowel, but I am sure it is used.
I am not sure if there is a low rounded central vowel.

Then we go to the ones we naturally round, /u,ʊ,o,ɔ/ and unrounded them.  The high two, /u,ʊ/ are unrounded to /ɯ/ (they don't seem to have a separate symbol for the tense and lax one). 
Then the other two, /o,ɔ/ become /ɤ/ (see above).  This is the <e> in pinyin, I believe.  I've only encountered this vowel a few times. 
The low rounded one seems to be /ɒ/.  It might be close to /a/ rounded but I'm not perfectly sure.

Now this might be daunting it's really not that hard.  But all these new vowels means we have tones of new diphthongs and tripthongs.  If anyone has any questions feel free to comment or message me.