Order Theory, Lattice Theory, or any directly relatable subject is not taught at my university, and quick searches don't give much clue into good textbooks for the subjects.
My question is: what are some good modern, introductory textbooks for order theory?
I have seen several questions regarding textbook recommendations for lattice theory (here and here). However, I was wanting something that dealt with both lattices and other orders (partial orders, preorders, etc) more generally. It may be that there isn't much to say about these more general orders when they aren't lattices, but I wouldn't know any better.
I wish I knew, but since you have references for lattice theory, here's one for everything but lattice theory: Ok, Real Analysis with Economics Applications. Start with Ch. B sec. 4. In general, there's a lot of material in economics about preorders as preferences are typically taken to be preorders.