Try process of elimination.
they’re
That apostrophe tells you two words have been combined. Remember that. Then think of what the two words are. Use “they’re” only when you mean “they are” and never at any other time.
That’s one down.
there
Relate “there” to “here” and “where.” They look like relatives. They are. Use them for the same kind of situation. You’re talking about place: something is here, something is over there; where is it?
When you’re building a sentence with “there is” (“There is a spot on your shirt”), it doesn’t really mean there (as if pointing with your finger) but is just an empty placeholder. Even then, you can sort of think of it slantwise with “here” or “where” (“Here is a spot on your shirt.” “Where is the spot on my shirt?”) and remember that it’s part of the same family. In a way it’s still about place.
Two down.
their
If you know you don’t mean “they are,” and you know you’re not talking about place (or placeholders), all that’s left is “their.” This one belongs to a family too, and you should just learn them all: the possessive pronouns. Here (or there) they are:
singular
my, mine
your/yours
his, her/ hers, its
plural
our/ours
your/yours
their/ theirs
The slashes mean alternative forms, depending: That is their house. The house is theirs.
They’re (they are) irregular. So are a lot of other things we just have to learn: one, two, three, I, myself, me, A, B, C.