Please note that German has long and short vowels. Usually the long vowels are before a single consonant or when the vowel is doubled, and the short are before a double consonant. This doesn't make a great deal of difference, especially for beginning speakers, but as an example, listen to the difference between Mann (man) and man (one).
A - As a short English A
Examples: das Abendessen the dinner; die Tasche the pocket/handbag
E - As English UH
Examples: die Kanne the teapot; das Regal the rack
I - As a long English E
Examples: das Geschirr the dishes; das Vitamin the vitamin
O - As a long English O
Examples: das Obst the fruit; die Dose the can
U - As English double-o in school
Examples: Die Schublade the drawer; der Hunger hunger
Ä (ASCII code 142), ä (ASCII code 132), some times written as Ae, ae
As a long English A
Examples: die Schränke the cupboards; die Gläser the glasses (not eye-glasses)
Ö (ASCII code 153), ö (ASCII code 148), some times written as Oe, oe
As English double-o in hook
Examples: der Dosenöffner the can-opener; die Öfen the ovens
Ü (ASCII code 154), ü (ASCII code 129), some times written as Ue, ue
As a German U with a tighter mouth
Examples: das Frühstück the breakfast; die Schüssel the bowl
AU
As English OW
Examples: der Waffelautomat the waffler; der Auflauf the pudding
EI
As a long English I
Examples: der Kürzzeitmesser the egg-timer; die Nachspeise the dessert
EU
As English OY
Examples: Deutsch german; heute to-day
IE
As a long English E
Examples: das Gefrierfach the freezer; der Korkenzieher the cork-screw
ÄU
As English OY
Examples: das Fräulein the miss; die Häuser the houses
Please not that the letters with an Umlaut (Ä, Ö, Ü) and the diphthongs are not separate letters, and are listed alphabetically as would be expected.