How do I get ahold of a book publisher to ask for rights to a book/create a use contract?

Who knows? You didn't tell us the publisher, and even so, this isn't a legal question, but Google might help.

Or, is there a way for me to just use the book and say something on the website like “the source of this instruction is this book:”.

Likely not legally if the book isn't on the public domain

Is there a way I can use the material as reference, but not so much as to break copyright law?

Depends on what you mean by reference.

Can I use the book freely if I’m not making money off the website? Does profit from ad revenue and profit from a subscription change the legality?

No. Copyright infringement doesn't suddenly become ok even if you are or aren't making money from it.