Order Theory and Lattice Theory Synonymous? Is Order Theory the same as Lattice Theory? Can anyone recommend good beginners text book on either?

I learned the basics from Introduction to Lattices and Order, 2nd Edition, by B. A. Davey and H. A. Priestley. Its not bad. To answer your title question: technically, order theory is more general than lattice theory. In practice you don't really study one without also studying the other.