First of all, I am not an attorney. This is not legal advice. I am not a literary agent. This is not business advice. These are my opinions and experience, presented as possible thinking guide. With luck, smarter people than I will contribute in comments to correct my errors and misstatements and present further wisdom.
New York novel contracts are much bigger than short story contracts in pretty much every sense of the term. Rule number one of a novel contract is to have an agent. If you don't have an agent, go get one. (This varies widely with independent press novel contracts, which tend to be much more modest and resemble short story contracts.)
However, even with an agent, you are responsible for your own career. No one cares more about your work than you do. That means you have to read and understand every damned word of those things.
If you have an eye for this sort of work, go find the appropriate SFWA model contract and read it. That will help.
Also, remember that while contracts are inherently adversarial (that's how our legal system works), they do not have to be inherently exploitative. The role of a contract is to preserve and protect the interests of both parties.
Know what it is your contract is doing. In a very loose sense, a literary contract is a license you grant your publisher to reproduce a copyright of which you are the owner. (This is different for work-for-hire, which is a whole 'nother animal about which I know even less than I know about original fiction. If you're doing work-for-hire, go research that very carefully.) Copyright licenses are divisible, but not in a statutory way. This means there is no legally defined meaning of "First rights", for example. There are certainly customary meanings, which are reflected in contracts, but the specific rights you are licensing to your publisher are always negotiable.
Note that in this sense, "negotiable" means they can say "no." So can you.
So look carefully at what rights you are licensing. English only, or foreign language rights? United States, North American, or world English rights? Those are pretty often subjects of negotiation.
Look even more carefully at derivative and subrights — electronic reproduction, film, television, gaming. Don't give those away to a print publisher ever, without a damned good reason that I can't imagine right now.
Make sure you understand exactly what you're licensing.
Look at the reversion clause. When does the book or story come back to you? If it comes back a certain length of time after a book is declared out of print, is "out of print" defined in your contract? What precisely does that mean?
Look at the exclusion clause. (As far as I know, this mostly applies to short fiction.) How soon can the story be resold? Are there exceptions for Year's Best or single-author collections? Is there a generic exception with written permission of the publisher.
In my case, I always look carefully at copyrights, and usually insert language about by-lines and promotional mentions. This is because my cover name, "Jay Lake", is not my legal name. There's nothing nefarious here, "Jay" is a nickname, but all my copyrights and contracts are held by some character named "Joseph E. Lake, Jr." (You can find him at the front of all my books.)
There's other fairly standard stuff, audit clauses and indemnification clauses and so on. Read and understand those.
As for payments, royalties, that sort of thing, I won't presume to give business advice here. It would be meaningless in the abstract if I did, and likely wrong in the specific even if I knew the specifics. That's what your agent is for.
Advice of counsel is a wonderful thing, but if you run your literary contracts past an attorney, you want to be sure you're dealing with an attorney who normally handles such things. A software intellectual property attorney, for example, will go nuts on a literary contract, waste a lot of billable time to you, and really irritate whoever you're negotiating with.
One last thing. Everything is negotiable, but that doesn't mean people will negotiate everything. You need to know what's important to the people you're negotiating with. For example, payment schedules might be flexible (cash flow) but royalty rates might not be (revenue). Understand what's reasonable, and act reasonable. I've never gone wrong trying to be nice about things, even when I have been irritated or upset.
Let the corrections and flaming begin in comments.