So I was pronouncing "gestern" with a "sht" sound like in "stadt", but after I heard the audio I did a quick research, it seems that "st" is only pronounced "sht" when the syllable starts with it.
So "Ver-ste-hen" = "sht" But "ges-tern" = "st"
Personally I would consider the syllables to be split this way "ge-stern", but I'm no linguist, and on forvo.com native speakers seem to pronounce it with a "st" sound anyway.
Initial 'st-' is always pronounced 'sht'. That's why it's pronounced 'sht' in compounds- it's technically an initial 'st', as you're jamming several words together. From your examples, I thought that we were only talking about medial 'st'- my mistake.
Don't forget to capitalise your nouns. :)
You shouldn't have too much trouble in identifying compound words with a little practice- the nouns are easy enough to figure out, and the verbs aren't too bad.
Verbs with separable prefixes are all prefixed with prepositions like gegen, aus, ab and mit, and verbs with inseparable prefixes have one of the following: be-, ent-, emp-, er-, ge-, miss-, ver- or zer-.
Best of luck!