A useful rule to follow is this:
は indicates that what precedes it is the topic of the whole of the rest of the sentence up to the next full stop
が marks the subject of the next verb that is in a sentence-ending form (and of any intervening verbs in non-sentence-ending forms) but not of any subsequent verbs in the same sentence.
In other words, は lasts for the whole sentence, but が is 'used up' by the next sentence-ending form, so any subsequent verb in the same sentence must have a different subject.
Try it out on these two:
こどもがねていますから "The child is asleep, so . . ." (こども is now used up, so the next verb will have a different subject)
しずかにしてください ". . . please be quiet" (subject is obviously the person addressed)
こどもはねていますから "The child is asleep, so . . ." (こども will be the topic/subject of the next verb so that verb can't be a request such as しずかにしてください: sentence 2 is therefore invalid. The sentence could perhaps conclude with something like ないていません, giving "The child is asleep, so it isn't crying")
The sentences traditionally used in Japanese school textbooks to illustrate this point are:
とりはなくときくちをあけます "When birds sing they open their mouths" (とり is the subject of both verbs: "Birds [topic] - when they sing they open their mouths"
とりがなくときくちをあけます "When birds sing I open my mouth" (Subject とり marked with が is used up by verb なく, so あけます must have a different subject, and in the absence of a new subject being provided we fall back on default "I".)
Consider also these examples:
ちちはびょうきになったから、いそいでかえった "My father, because he fell ill, hurried home"
ちちがびょうきになったから、いそいでかえった "Because my father fell ill, I hurried home"
おとうとはらくだいしたからだいがくをやめた "My younger brother failed his exams so he dropped out of university"
おとうとがらくだいしたからだいがくをやめた "My younger brother failed his exams, so I dropped out of university