Question: The domain is the set of all words in the English language (as defined by, say, Webster's dictionary). Word $x$ is related to word $y$ if $x$ appears as a substring of y. For example, "ion" is related to the word "companions" because the letters i-o-n appear in order in the word "companions". Is it a partial order, strict order, and is it a total order?
My thoughts: It is reflexive because a $x$ being a substring of $y$ makes it a substring of itself. It is transitive because if it is a substring of $y$ and $y$ is a substring of $z$ then it is a substring of $z$. It is antisymmetric because if $x$ is a substring of $y$ and $y$ is a substring of $x$ then they are the same.
Am I doing this right?
Yes, you're doing it right, however you have to use the fact that words are finite in order to prove that this is antisymmetric. Consider the infinite case, the word $abababab\ldots$ then it is a substring of the word $babababa\ldots$ and vice versa, but they are not equal.
So it is a partial order. The other two do not hold, but I'll leave it to you to figure out why.