Look, I know as well as you do that all of those words up there in the title sound the damn same. And I know it’s completely awful and unfair. They’re called heterographs – words that are spelled differently but pronounced the same way, in case you care. To many people, they are simply The Enemy.
It would certainly be easiest just need to memorize when to use each word, but the truth is that many people simply cannot do so. I feel you – the older I get, the harder I find it to even remember a new phone number. So, as with many of English’s quirks and foibles, we turn to visual tricks and reminders for assistance.
“They’re” is the simplest, I think: the apostrophe’s placement makes a sort of triangular shape between the y and the r, implying the missing “A.” If you look for it, and then use the same trick as I suggested for “its” and “it’s,” reading the sentence aloud and then switching it to “they are,” you can at least eliminate “they’re” when it’s clearly incorrect.
Unfortunately, that still leaves you with two choices. One way to make it a little easier, though it might seem childish, is to think of the lowercase “i” in the word “their” as a little stick figure person, with the dot representing the head. If what you want to say is that a thing or situation belongs to or pertains to a person, animal, or entity (such as a business, e.g. “Target closed their stores in Canada last year”), you can imagine a tiny person in the word, and use “their.” If that’s not the case, you will usually be safe in defaulting to “there.”
“There” is the most ambiguous one of this evil triad, since it can be used before a verb (there is, there goes, etc.), as a substitution for a place (Who’s in there?), and even as an expression of comfort or sympathy (there, there). Therefore (pun intended), if we can eliminate the other two heterographs as shown above, we can significantly increase the possibility that “there” is the correct word to use.
As always, I have the utmost sympathy for people who learn English as a second language, particularly later in life. Adult brains are not the sponges that children’s brains are; they may need more help remembering and synthesizing language. When spell checkers are no help (as in this situation), it can be incredibly valuable to have memory triggers and tests like these to help you out. It can even be fun: draw little faces on your “theirs,” do a dramatic reading of sentences where “they are” is clearly the wrong answer; write crazy sentences that use “there” in all of its possible permutations. I believe in taking writing seriously, but that doesn’t mean there’s no room for people to have their fun, as long as they’re learning in the process.