I was recently rereading bits of Pride and Prejudice and the dialog is absolutely brilliant. How can I learn to write dialog that sounds like it is from this general era? For example are there any lists of words that I should try to substitute or differences in the grammatical constructs used?
Your best bet is to read a lot of the literature written in that time period. Those authors have an advantage over you - they lived then and knew how people spoke. When writing a historical piece an etymology dictionary is your best friend. It'll tell you if the word you want to use was in use back then and if not, when it came into usage. But you're going to want to do a lot of research into classes in that time period. An upperclassman wouldn't speak the same as a lowerclassman. But reading a lot of books from that era can help you get a pretty good feel for things. Literature can often be the best forms of history.
I wouldn't try to imitate exactly the style of dialogue in the era of Austen unless you are planning on imitating everything about the historical era very well. As Ralph Gallagher said, Austen and others have an advantage over you - they were just writing how they spoke or heard others around them speak. If you manage to only get part of the whole historical ethos - some of the words and a bit of the dress with anachronisms from our own time thrown in - your readers will be continuously jilted out of the story.
It is possible to write historical fiction using the languages of our own day. No one would write a work of historical fiction set in Chaucer's time and attempt to use his language because readers would largely not understand it. Instead the language would be modernized. If you can't do the Romantic era perfectly, you might be best off doing the same.
It is possible for dialogue written in modern English or any language to be brilliant. What matters is an ear for cadences, for apt words, for truths about life as they really are. Whether you write in Middle, Romantic, Victorian, or modern English, it's what you say that matters.
Do historical dialogue extremely well or not at all.
Read scenes out loud, learn to speak like that. A key part of writing dialogue is being able to hear it back in your head like people actually speaking to one another. It's a skill George Lucas never bothered to acquire. They call it having an "ear" for dialogue for a reason.