"There", "their" and "they're" are often confused for each other, but by process of elimination, choosing the correct word is simple. Just follow these steps.
Write out the complete sentence that you are working to figure out the grammar rule for.
Leave a space or an underline where either "there", "their", or "they're" belongs.
Read the sentence out loud replacing the space with the phrase "they are". "They're" is the abbreviation for "they are" and if replacing the space made sense you have figured out which of the confused words to use. If "they are" did not make sense, then you have eliminated one of the three options.
Read the sentence again replacing the space with the word "her". You can do this because "their" and "her" are both possessive forms. If using "her" made sense as a replacement, then you can fill the space with "their".
Fill the blank space with "there" if the other two choices did not work. "There" can indicate space and also begin an explanation.
Be cautious of "theirs" and "there's". Try saying "there is" in the space, which is the long form of "there's". Use the same process of elimination to find the correct commonly confused word.