The German language may be known for its many consonants, but what most learners have trouble with are in fact the vowels.
The following chart gives a pretty much thorough account of the pronunciation of German vowels, in form of a flowchart. It will help you determine how to pronounce the vowels in a given word.
UPDATE: This is version 5 of the chart, as of 2017-02-17.
Please comment here with suggestions, corrections, etc.