Start with the German alphabet. Sometimes A, O and U are umlauted, which means they have two little dots above them---Ä, -- and Ü. This changes their pronunciation. Pronounce the short "ä" like the "e" in "get". You'll see this in words like "Männer", the German word for "men". Pronounce the "ö" in German with a bit of an "er" sound. For example, to say "would like"---"möchten" leans more toward "merkten" in sound. Squeeze your lips into a tight "o" formation, but try to say the "er" sound in order to pronounce the "ü" in German. Many native English speakers find this sound hard to distinguish and pronounce.
Use the "sh" sound in English to pronounce an "s" at the beginning of a word. For example, the German word for mirror---"Spiegel"---is pronounced "shpeegel."
Pronounce "ch" at the beginning of a word like "sh" in English. "Sch" is also pronounced like the English "sh". However, the letter combination of "tsch" in German sounds like "ch" in English.
Remember that "e" in German sounds like a short "a" in English. Thus "Porsche" sounds like Porscha. Additionally, it's never a silent letter as it can be in English.
Go against what you might think is correct when saying the "ie"/ "ei" combination. In German the second letter of the two is the one pronounced, and it's long. Therefore, the "ie" in "Spiegel" is said like "ee", whereas the "ei" in die "Gesundheit" rhymes with the "i" in kite.
Consider the "j" in German the equivalent of "y" in English. For example, the German word for jacket--"Jacke"--sounds sort of like "yahcka."
Think of the letter "r" at the end of a German word as if it were a vowel. For example, the word "Vater" sounds closer to "Fata". The hard American "r" isn't spoken out in German. Roll "r's" in the middle of a word ever so slightly.
Pronounce "w" as if it's a "v" in English and "v" as if it's an "f". For example, Volkswagen is actually pronounced "folksvagen."
Know that the letter "y" in German is a bit of a wildcard. Depending upon where it is in the sentence, it can have a bit of a "oo" sound as in the word "typisch", an "ee" sound as in "Hobby" or a short "i" sound as in "Hymne"
Pronounce the "z" like the two "z's" in pizza.
Say the "eu" or "äu" combination as "oy," and the "au" (without the umlaut), such as in "Bauer," like "bow wow."