Thank you for including the screenshots, which clued me that you are on the old v1.9 version of Scrivener. Next time you post, be sure to mention you’re on v1.9, as people assuming you’re on v3.0 will give you all kinds of confusing advice with features and/or menu options that your version doesn’t have. Particularly in the case of your current question, as compile has changed drastically between the versions. 
Yes, you are absolutely on the right track, and by the time I type this up you will probably have already figured it out. 
Your screenshot is showing the Formatting Pane. You just need to add another Folder Level. Try this:
In the Formatting Pane:
- Click on folder Level 1+
- Click on the + (plus sign) to the right of Options, in the upper right corner.
That will add a new folder Level 2+.
In my screenshot below, when I select folder Level 2+, see how the Binder items for Chapters are highlighted in yellow? That’s how you know folder Level 2+ corresponds to Chapters.
Same thing with folder Level 1+, which corresponds to Parts.
Now what you need to do is set up your Part & Chapter page formatting.
Again, in the Formatting Pane:
- Click on folder Level 1+
- Click on the Section Layout button.
You’ll see something like this:
Change the contents of the Prefix and Suffix boxes to how you want your Parts to print.
Then Click on folder Level 2+ and do the same for Chapters.
I’m no expert on compile, so I might have missed something. Come back with questions if you get stuck.
See section 23.8 Formatting in the user manual for more details. Also, the interactive tutorial has a short but sweet overview of Compile formatting, in Step 16: Compiling the Draft.
The other thing you might want to do is create a new test project, using the “Novel (with Parts)” template. I’m suggesting this because Novel (with Parts) is set up to compile with the same structure as your project, so reviewing it might give you some tips. That’s what I did to get the screenshots in my examples.
Best,
Jim
