Should request payment details for a project I am to do?
Compensation is an integral part of the decision to take on any job. Not only should you know the amount ahead of time, but also the mode of compensation: is it 1k up front? Is it $10/hour? Is it $500 now and $500 at the end of the project? This is information that you are owed as a prospective employee.

You should just be straightforward about this. There’s no decorum that you need to follow outside of whatever is usual in your relationship. I would write something along these lines:


  Hi James,
  
  I’m excited about the research opportunity that you mentioned after class last week. I was hoping you could fill me in on some concrete details such as when and how much I will be paid, the expected project timeline, the scope of my part of the project, and who else would be involved.
  
  If you don’t have these details set yet, I totally understand. In that case, do you have an idea of when you will have them?
  
  Thanks,
  Stella


I’ve sent some variant of this email tons of times. I’m somewhat making up the details, as I don’t know what you have or have no discusses with this professor, but I would strongly recommend that you ask about any of the things that I have listed that you don’t know already. If this is a project that is part of publishable research, you should also ask about authorship and (if applicable to your field) authorship order.