I'm glad that helped:
> I keep thinking in terms of specific examples...
If you do that, which can be dangerous, you need enough examples to keep from getting led astray by the peculiarities of special cases.
In particular, you need to pay special attention to "silly" examples, where the structures are chosen in a "trivial" way. The preorder where everything is \\(\le\\) to everything else; the preorder where anything is only \\(\le\\) itself - all other examples stand somewhere between these two.
Another important thing to do is this: whenever someone defines some sort of gadget, like a "preorder", and then defines a nice special case with extra features, like a "poset", you need to look for examples of posets that _aren't_ preorders. Without this, you can't see what the special extra features are buying you! Mathematicians automatically do this whenever they learn new concepts.
It's just like if you were a biologist and you'd just learned about "animals" and "vertebrates": you'd have to learn about some animals that aren't vertebrates.