Hi~
 

This is Teacher Rach~ Here are your corrections:

My younger sister asked me to look her cat while her family go for a trip yesterday.

==> My younger sister asked me to look after her cat while her family went on a trip yesterday.

She told me how to take care of her cat over and over.

So I said I know exactly how you feel. ^^

Well done~! Your errors are very minimal. Please take note of these:

*Vocabulary: You should use "look after" if you mean "to take care of something or someone".

*verb tense:  Use the simple past tense of "go" since the action was "yesterday".

I hope the suggestions will help! Keep posting~~ ^^