For a proper pronounce you should know three things:
Iotizing means that if Е stands at the beginning of the word, or after another vowel, or after hard or soft sign (Ь or Ъ), then it's iotized, i.e. it produces two sounds: Й+Э (as in YEllow)
Softening(Palatalisation) means that if Е stands after a consonant, then the consonant becomes a "soft variant"("palatalised"). (Note that Е gives only one sound "Э", as in Elbow, in this case). But if the consonant has no "soft variant"[Ш, Ж, Ц], then it doesn't change, of course.
Ель -> ЙЭл';
ЛЕc -> Л'Эс;
ЖЕчь -> ЖЭч';
ШЕст -> ШЭст etc.
Still one step remains so far. It's the Reduction.
Reduction may be applied to unstressed vowel (after proper iotizing and softening!) to pronounce it differently, so this is a matter of a dialect. Usually, unstressed Russian "Е" gives "И" (this is the most standard way, called "Иканье"). Though saying just "Э" is OK too.
So "Е(катеринбург)" usually yields "ЙИ(кат'ир'инбурк)". But stressed vowel is never changed, e.g.: подъЕзд -> падЙЭст.