Listen to the vowels a, e, i, o, and u by themselves, either from a native speaker you know pronounces them, or by listening to them online.
Search the Internet for "German vowels and audio" to find interactive audio examples of how to pronounce German vowels. A benefit of listening to them online is that you can also rewind and hear the pronunciation again, as well as pause and repeat. This can also be done with beginning German language course material. You can purchase such a course online, from a book store, or you can check one out from you local public library.
YouTube.com also has some German language videos. In these videos you can see the words being pronounced by speakers in many cases, which will help you form the vowels correctly. Search for "German and vowels" to find these videos.
Focus on learning the pronunciation of the umlaut next. An umlaut is the diacritical mark above the letters a, o, and u in German. It changes the pronunciation of the vowel so that it "is spoken from the front of the mouth. It uses the tongue in a curling action to produce the sound needed," according to LearnGermanWords.com.
Practice the pronunciation of each vowel and umlaut in different words that use the vowel. German vowels each have their own, clear sound. Focus on producing each sound precisely as you practice saying vowels and umlauts in words.