I believe you can only draw notes and use the extra features (bulleted lists, etc) on notes that exist in the "On my iPhone" / "On my iPad" folder within the Notes app (I could be mistaken but this is how I'm seeing it). To access and use notes "On my iPhone" / "On my iPad":
1. Open the Notes app
2. At the top of the screen, tap the back arrow
3. Find "On my iPhone" / "On my iPad" in the list
4. Tap on "Notes" under the "On my iPhone" / "On my iPad" heading
5. Tap on the new note icon at the bottom-right (iPhone) or top-right (iPad)
You can now tap the round + button which will reveal the additional features (iPhone)
Or tap on the new features just above the keyboard on the right side (iPad)
I don't know if the new features can be added to old notes that already exist on the device unless you move those notes to the "On my iPhone" / "On my iPad" notes folder. I also did NOT update whatever the Notes app was wanting to update when it was first opened immediately after the iOS 9 update (I now forget what that even was).