As an online teacher of English for Chinese students, I have found that vowel sounds hold an important key to helping students with pronunciation. Students, of course, need to practice beginning and ending sounds, but the key, at least for my Chinese students, is learning the correct long and short vowel sounds.
I know that there are other pronunciation errors that come up. One common error is that the student may add the /uh/ sound after the beginning sound (mu), or at the end of the word (and uh). They also have difficulty with digraphs (th, sh, ch), but if I could spend an extra ten minutes on pronunciation, I would spend eight of those minutes on vowel sounds. That is how important of a role I think they play in achieving correct and clear pronunciation.
It is kind of funny, the other day I was watching the movie, "My Fair Lady," with Audrey Hepburn. She played the role of a poor flower girl who was taken in as part of a bet by two linguists. The bet was that she would be able to be passed off as an educated shopgirl. As part of this process, she had to learn to speak proper English. She could not get that long /a/ sound. It took a lot of practice and then one day it just clicked. That is how I think it works for students. They need that practice. But, the practice needs to be done correctly, not just repeating them incorrectly. This is why I think that practicing the vowel sounds in isolation is one of the first steps.
When I teach a lesson, I let the students read the words. I use this time to listen to their pronunciation. If I hear that they are pronouncing a vowel sound incorrectly, I take the time to focus on that one vowel sound. Then I have the student read the words again. As they read, I will nod and give them a thumbs up to indicate they are saying the words correctly.
If a student is really struggling with vowel sounds, I spend some time at the end of the lesson focusing on these sounds. I show them the vowels, which I have on a card. I explain that the long vowels say their letter name, which my Chinese students usually get pretty quickly. Then I spend some time on the short vowels.